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G = C22×C23.D7  order 448 = 26·7

Direct product of C22 and C23.D7

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×C23.D7, C25.2D7, C24⋊4Dic7, C24.79D14, (C23×C14)⋊8C4, (C24×C14).4C2, C23⋊5(C2×Dic7), C14.52(C23×C4), (C2×Dic7)⋊9C23, (C2×C14).321C24, (C23×Dic7)⋊10C2, (C22×C14).212D4, C14.175(C22×D4), C2.14(C23×Dic7), C22⋊2(C22×Dic7), C22.50(C23×D7), C23.108(C7⋊D4), C23.343(C22×D7), (C23×C14).112C22, (C22×C14).428C23, (C22×Dic7)⋊50C22, C14⋊3(C2×C22⋊C4), C7⋊3(C22×C22⋊C4), (C2×C14)⋊9(C22×C4), (C2×C14)⋊9(C22⋊C4), C2.4(C22×C7⋊D4), (C22×C14)⋊17(C2×C4), (C2×C14).592(C2×D4), C22.121(C2×C7⋊D4), SmallGroup(448,1292)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C22×C23.D7
Chief series
C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C23×Dic7 — C22×C23.D7
Lower central
C7 — C14 — C22×C23.D7
Upper central
C1 — C24 — C25

Generators and relations for C22×C23.D7
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f7=1, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, gcg-1=ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg-1=f-1 >

Subgroups: 1796 in 674 conjugacy classes, 287 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C24, C24, C24, Dic7, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C23×C4, C25, C2×Dic7, C2×Dic7, C22×C14, C22×C14, C22×C22⋊C4, C23.D7, C22×Dic7, C22×Dic7, C23×C14, C23×C14, C23×C14, C2×C23.D7, C23×Dic7, C24×C14, C22×C23.D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, C24, Dic7, D14, C2×C22⋊C4, C23×C4, C22×D4, C2×Dic7, C7⋊D4, C22×D7, C22×C22⋊C4, C23.D7, C22×Dic7, C2×C7⋊D4, C23×D7, C2×C23.D7, C23×Dic7, C22×C7⋊D4, C22×C23.D7

Smallest permutation representation of C22×C23.D7
►On 224 points
Generators in S224
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)(113 176)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 169)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 190)(128 191)(129 192)(130 193)(131 194)(132 195)(133 196)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 204)(142 205)(143 206)(144 207)(145 208)(146 209)(147 210)(148 197)(149 198)(150 199)(151 200)(152 201)(153 202)(154 203)(155 218)(156 219)(157 220)(158 221)(159 222)(160 223)(161 224)(162 211)(163 212)(164 213)(165 214)(166 215)(167 216)(168 217)

(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 161)(120 162)(121 163)(122 164)(123 165)(124 166)(125 167)(126 168)(127 141)(128 142)(129 143)(130 144)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 151)(138 152)(139 153)(140 154)(169 211)(170 212)(171 213)(172 214)(173 215)(174 216)(175 217)(176 218)(177 219)(178 220)(179 221)(180 222)(181 223)(182 224)(183 197)(184 198)(185 199)(186 200)(187 201)(188 202)(189 203)(190 204)(191 205)(192 206)(193 207)(194 208)(195 209)(196 210)

(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(113 204)(114 205)(115 206)(116 207)(117 208)(118 209)(119 210)(120 197)(121 198)(122 199)(123 200)(124 201)(125 202)(126 203)(127 218)(128 219)(129 220)(130 221)(131 222)(132 223)(133 224)(134 211)(135 212)(136 213)(137 214)(138 215)(139 216)(140 217)(141 176)(142 177)(143 178)(144 179)(145 180)(146 181)(147 182)(148 169)(149 170)(150 171)(151 172)(152 173)(153 174)(154 175)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 196)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)

(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)(141 162)(142 163)(143 164)(144 165)(145 166)(146 167)(147 168)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)(169 190)(170 191)(171 192)(172 193)(173 194)(174 195)(175 196)(176 183)(177 184)(178 185)(179 186)(180 187)(181 188)(182 189)(197 218)(198 219)(199 220)(200 221)(201 222)(202 223)(203 224)(204 211)(205 212)(206 213)(207 214)(208 215)(209 216)(210 217)

(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)

(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)

(1 138 22 117)(2 137 23 116)(3 136 24 115)(4 135 25 114)(5 134 26 113)(6 140 27 119)(7 139 28 118)(8 131 15 124)(9 130 16 123)(10 129 17 122)(11 128 18 121)(12 127 19 120)(13 133 20 126)(14 132 21 125)(29 166 50 145)(30 165 51 144)(31 164 52 143)(32 163 53 142)(33 162 54 141)(34 168 55 147)(35 167 56 146)(36 159 43 152)(37 158 44 151)(38 157 45 150)(39 156 46 149)(40 155 47 148)(41 161 48 154)(42 160 49 153)(57 194 78 173)(58 193 79 172)(59 192 80 171)(60 191 81 170)(61 190 82 169)(62 196 83 175)(63 195 84 174)(64 187 71 180)(65 186 72 179)(66 185 73 178)(67 184 74 177)(68 183 75 176)(69 189 76 182)(70 188 77 181)(85 222 106 201)(86 221 107 200)(87 220 108 199)(88 219 109 198)(89 218 110 197)(90 224 111 203)(91 223 112 202)(92 215 99 208)(93 214 100 207)(94 213 101 206)(95 212 102 205)(96 211 103 204)(97 217 104 210)(98 216 105 209)
 
G:=sub<Sym(224)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,197)(149,198)(150,199)(151,200)(152,201)(153,202)(154,203)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)(161,224)(162,211)(163,212)(164,213)(165,214)(166,215)(167,216)(168,217), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222)(181,223)(182,224)(183,197)(184,198)(185,199)(186,200)(187,201)(188,202)(189,203)(190,204)(191,205)(192,206)(193,207)(194,208)(195,209)(196,210), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(113,204)(114,205)(115,206)(116,207)(117,208)(118,209)(119,210)(120,197)(121,198)(122,199)(123,200)(124,201)(125,202)(126,203)(127,218)(128,219)(129,220)(130,221)(131,222)(132,223)(133,224)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,176)(142,177)(143,178)(144,179)(145,180)(146,181)(147,182)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)(161,196)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(169,190)(170,191)(171,192)(172,193)(173,194)(174,195)(175,196)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,138,22,117)(2,137,23,116)(3,136,24,115)(4,135,25,114)(5,134,26,113)(6,140,27,119)(7,139,28,118)(8,131,15,124)(9,130,16,123)(10,129,17,122)(11,128,18,121)(12,127,19,120)(13,133,20,126)(14,132,21,125)(29,166,50,145)(30,165,51,144)(31,164,52,143)(32,163,53,142)(33,162,54,141)(34,168,55,147)(35,167,56,146)(36,159,43,152)(37,158,44,151)(38,157,45,150)(39,156,46,149)(40,155,47,148)(41,161,48,154)(42,160,49,153)(57,194,78,173)(58,193,79,172)(59,192,80,171)(60,191,81,170)(61,190,82,169)(62,196,83,175)(63,195,84,174)(64,187,71,180)(65,186,72,179)(66,185,73,178)(67,184,74,177)(68,183,75,176)(69,189,76,182)(70,188,77,181)(85,222,106,201)(86,221,107,200)(87,220,108,199)(88,219,109,198)(89,218,110,197)(90,224,111,203)(91,223,112,202)(92,215,99,208)(93,214,100,207)(94,213,101,206)(95,212,102,205)(96,211,103,204)(97,217,104,210)(98,216,105,209)>;
 
G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,197)(149,198)(150,199)(151,200)(152,201)(153,202)(154,203)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)(161,224)(162,211)(163,212)(164,213)(165,214)(166,215)(167,216)(168,217), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222)(181,223)(182,224)(183,197)(184,198)(185,199)(186,200)(187,201)(188,202)(189,203)(190,204)(191,205)(192,206)(193,207)(194,208)(195,209)(196,210), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(113,204)(114,205)(115,206)(116,207)(117,208)(118,209)(119,210)(120,197)(121,198)(122,199)(123,200)(124,201)(125,202)(126,203)(127,218)(128,219)(129,220)(130,221)(131,222)(132,223)(133,224)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,176)(142,177)(143,178)(144,179)(145,180)(146,181)(147,182)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)(161,196)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(169,190)(170,191)(171,192)(172,193)(173,194)(174,195)(175,196)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,138,22,117)(2,137,23,116)(3,136,24,115)(4,135,25,114)(5,134,26,113)(6,140,27,119)(7,139,28,118)(8,131,15,124)(9,130,16,123)(10,129,17,122)(11,128,18,121)(12,127,19,120)(13,133,20,126)(14,132,21,125)(29,166,50,145)(30,165,51,144)(31,164,52,143)(32,163,53,142)(33,162,54,141)(34,168,55,147)(35,167,56,146)(36,159,43,152)(37,158,44,151)(38,157,45,150)(39,156,46,149)(40,155,47,148)(41,161,48,154)(42,160,49,153)(57,194,78,173)(58,193,79,172)(59,192,80,171)(60,191,81,170)(61,190,82,169)(62,196,83,175)(63,195,84,174)(64,187,71,180)(65,186,72,179)(66,185,73,178)(67,184,74,177)(68,183,75,176)(69,189,76,182)(70,188,77,181)(85,222,106,201)(86,221,107,200)(87,220,108,199)(88,219,109,198)(89,218,110,197)(90,224,111,203)(91,223,112,202)(92,215,99,208)(93,214,100,207)(94,213,101,206)(95,212,102,205)(96,211,103,204)(97,217,104,210)(98,216,105,209) );
 
G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,105),(113,176),(114,177),(115,178),(116,179),(117,180),(118,181),(119,182),(120,169),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(127,190),(128,191),(129,192),(130,193),(131,194),(132,195),(133,196),(134,183),(135,184),(136,185),(137,186),(138,187),(139,188),(140,189),(141,204),(142,205),(143,206),(144,207),(145,208),(146,209),(147,210),(148,197),(149,198),(150,199),(151,200),(152,201),(153,202),(154,203),(155,218),(156,219),(157,220),(158,221),(159,222),(160,223),(161,224),(162,211),(163,212),(164,213),(165,214),(166,215),(167,216),(168,217)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(113,155),(114,156),(115,157),(116,158),(117,159),(118,160),(119,161),(120,162),(121,163),(122,164),(123,165),(124,166),(125,167),(126,168),(127,141),(128,142),(129,143),(130,144),(131,145),(132,146),(133,147),(134,148),(135,149),(136,150),(137,151),(138,152),(139,153),(140,154),(169,211),(170,212),(171,213),(172,214),(173,215),(174,216),(175,217),(176,218),(177,219),(178,220),(179,221),(180,222),(181,223),(182,224),(183,197),(184,198),(185,199),(186,200),(187,201),(188,202),(189,203),(190,204),(191,205),(192,206),(193,207),(194,208),(195,209),(196,210)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(113,204),(114,205),(115,206),(116,207),(117,208),(118,209),(119,210),(120,197),(121,198),(122,199),(123,200),(124,201),(125,202),(126,203),(127,218),(128,219),(129,220),(130,221),(131,222),(132,223),(133,224),(134,211),(135,212),(136,213),(137,214),(138,215),(139,216),(140,217),(141,176),(142,177),(143,178),(144,179),(145,180),(146,181),(147,182),(148,169),(149,170),(150,171),(151,172),(152,173),(153,174),(154,175),(155,190),(156,191),(157,192),(158,193),(159,194),(160,195),(161,196),(162,183),(163,184),(164,185),(165,186),(166,187),(167,188),(168,189)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133),(141,162),(142,163),(143,164),(144,165),(145,166),(146,167),(147,168),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161),(169,190),(170,191),(171,192),(172,193),(173,194),(174,195),(175,196),(176,183),(177,184),(178,185),(179,186),(180,187),(181,188),(182,189),(197,218),(198,219),(199,220),(200,221),(201,222),(202,223),(203,224),(204,211),(205,212),(206,213),(207,214),(208,215),(209,216),(210,217)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,138,22,117),(2,137,23,116),(3,136,24,115),(4,135,25,114),(5,134,26,113),(6,140,27,119),(7,139,28,118),(8,131,15,124),(9,130,16,123),(10,129,17,122),(11,128,18,121),(12,127,19,120),(13,133,20,126),(14,132,21,125),(29,166,50,145),(30,165,51,144),(31,164,52,143),(32,163,53,142),(33,162,54,141),(34,168,55,147),(35,167,56,146),(36,159,43,152),(37,158,44,151),(38,157,45,150),(39,156,46,149),(40,155,47,148),(41,161,48,154),(42,160,49,153),(57,194,78,173),(58,193,79,172),(59,192,80,171),(60,191,81,170),(61,190,82,169),(62,196,83,175),(63,195,84,174),(64,187,71,180),(65,186,72,179),(66,185,73,178),(67,184,74,177),(68,183,75,176),(69,189,76,182),(70,188,77,181),(85,222,106,201),(86,221,107,200),(87,220,108,199),(88,219,109,198),(89,218,110,197),(90,224,111,203),(91,223,112,202),(92,215,99,208),(93,214,100,207),(94,213,101,206),(95,212,102,205),(96,211,103,204),(97,217,104,210),(98,216,105,209)]])
 

136 conjugacy classes

class 1 2A···2O2P···2W4A···4P7A7B7C14A···14CO
order12···22···24···477714···14
size11···12···214···142222···2

136 irreducible representations

dim1111122222
type++++++-+
imageC1C2C2C2C4D4D7Dic7D14C7⋊D4
kernelC22×C23.D7C2×C23.D7C23×Dic7C24×C14C23×C14C22×C14C25C24C24C23
# reps112211683242148

Matrix representation of C22×C23.D7 ►in GL6(𝔽29)

100000
0280000
001000
000100
0000280
0000028
,
2800000
010000
0028000
0002800
000010
000001
,
100000
010000
001000
00252800
0000280
000001
,
100000
010000
0028000
0002800
000010
000001
,
100000
010000
0028000
0002800
0000280
0000028
,
100000
010000
001000
000100
0000160
0000020
,
2800000
0280000
002100
00242700
0000020
0000160

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,25,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,20],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,2,24,0,0,0,0,1,27,0,0,0,0,0,0,0,16,0,0,0,0,20,0] >;
 

C22×C23.D7 in GAP, Magma, Sage, TeX 

C_2^2\times C_2^3.D_7
 
% in TeX
 
G:=Group("C2^2xC2^3.D7");
 
// GroupNames label
 
G:=SmallGroup(448,1292);
 
// by ID
 
G=gap.SmallGroup(448,1292);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,1123,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^7=1,g^2=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*c*g^-1=c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,g*f*g^-1=f^-1>;
 
// generators/relations
 

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