Copied to
clipboard

G = C2×C23.D14  order 448 = 26·7

Direct product of C2 and C23.D14

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

Aliases: C2×C23.D14, C24.22D14, (C2×C14).26C24, C4⋊Dic7⋊50C22, C22⋊C4.85D14, C14⋊2(C42⋊2C2), (C2×C28).126C23, Dic7⋊C4⋊47C22, (C4×Dic7)⋊72C22, (C22×C4).168D14, C23.78(C22×D7), C22.68(C23×D7), C22.71(C4○D28), (C23×C14).52C22, C23.D7.84C22, C22.65(D4⋊2D7), (C22×C14).118C23, (C22×C28).350C22, (C2×Dic7).176C23, (C22×Dic7).203C22, C7⋊2(C2×C42⋊2C2), (C2×C4×Dic7)⋊29C2, (C2×C4⋊Dic7)⋊18C2, C14.11(C2×C4○D4), C2.13(C2×C4○D28), C2.8(C2×D4⋊2D7), (C2×Dic7⋊C4)⋊34C2, (C2×C22⋊C4).17D7, (C14×C22⋊C4).20C2, (C2×C4).255(C22×D7), (C2×C23.D7).21C2, (C2×C14).100(C4○D4), (C7×C22⋊C4).108C22, SmallGroup(448,935)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C2×C23.D14
Chief series
C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×C4×Dic7 — C2×C23.D14
Lower central
C7 — C2×C14 — C2×C23.D14
Upper central
C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C23.D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 852 in 246 conjugacy classes, 111 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C2×C42⋊2C2, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C22⋊C4, C22×Dic7, C22×C28, C23×C14, C23.D14, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×C23.D7, C14×C22⋊C4, C2×C23.D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C42⋊2C2, C2×C4○D4, C22×D7, C2×C42⋊2C2, C4○D28, D4⋊2D7, C23×D7, C23.D14, C2×C4○D28, C2×D4⋊2D7, C2×C23.D14

Smallest permutation representation of C2×C23.D14
►On 224 points
Generators in S224
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 197)(30 198)(31 199)(32 200)(33 201)(34 202)(35 203)(36 204)(37 205)(38 206)(39 207)(40 208)(41 209)(42 210)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(50 218)(51 219)(52 220)(53 221)(54 222)(55 223)(56 224)(85 136)(86 137)(87 138)(88 139)(89 140)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(141 192)(142 193)(143 194)(144 195)(145 196)(146 169)(147 170)(148 171)(149 172)(150 173)(151 174)(152 175)(153 176)(154 177)(155 178)(156 179)(157 180)(158 181)(159 182)(160 183)(161 184)(162 185)(163 186)(164 187)(165 188)(166 189)(167 190)(168 191)

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

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

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

(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)

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

88 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N4O4P4Q4R7A7B7C14A···14U14V···14AG28A···28X
order12···2224444444···4444477714···1414···1428···28
size11···14422224414···14282828282222···24···44···4

88 irreducible representations

dim11111112222224
type+++++++++++-
imageC1C2C2C2C2C2C2D7C4○D4D14D14D14C4○D28D4⋊2D7
kernelC2×C23.D14C23.D14C2×C4×Dic7C2×Dic7⋊C4C2×C4⋊Dic7C2×C23.D7C14×C22⋊C4C2×C22⋊C4C2×C14C22⋊C4C22×C4C24C22C22
# reps181212131212632412

Matrix representation of C2×C23.D14 ►in GL6(𝔽29)

100000
010000
0028000
0002800
0000280
0000028
,
100000
7280000
0028000
0002800
000010
00001228
,
2800000
0280000
001000
000100
0000280
0000028
,
100000
010000
001000
000100
0000280
0000028
,
1700000
0170000
0021800
00212600
0000172
0000012
,
1540000
16140000
0021100
0052700
0000124
0000028

G:=sub<GL(6,GF(29))| [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],[1,7,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,12,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,21,21,0,0,0,0,8,26,0,0,0,0,0,0,17,0,0,0,0,0,2,12],[15,16,0,0,0,0,4,14,0,0,0,0,0,0,2,5,0,0,0,0,11,27,0,0,0,0,0,0,1,0,0,0,0,0,24,28] >;
 

C2×C23.D14 in GAP, Magma, Sage, TeX 

C_2\times C_2^3.D_{14}
 
% in TeX
 
G:=Group("C2xC2^3.D14");
 
// GroupNames label
 
G:=SmallGroup(448,935);
 
// by ID
 
G=gap.SmallGroup(448,935);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,100,1571,297,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=c,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^13>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁