Copied to
clipboard

## G = C22×D4.D7order 448 = 26·7

### Direct product of C22 and D4.D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C22×D4.D7
 Chief series C1 — C7 — C14 — C28 — Dic14 — C2×Dic14 — C22×Dic14 — C22×D4.D7
 Lower central C7 — C14 — C28 — C22×D4.D7
 Upper central C1 — C23 — C22×C4 — C22×D4

Generators and relations for C22×D4.D7
G = < a,b,c,d,e,f | a2=b2=c4=d2=e7=1, f2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf-1=c-1, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 1044 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C14, C14, C14, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, C28, C2×C14, C2×C14, C22×C8, C2×SD16, C22×D4, C22×Q8, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C7×D4, C7×D4, C22×C14, C22×C14, C22×SD16, C2×C7⋊C8, D4.D7, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, D4×C14, D4×C14, C23×C14, C22×C7⋊C8, C2×D4.D7, C22×Dic14, D4×C2×C14, C22×D4.D7
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C24, D14, C2×SD16, C22×D4, C7⋊D4, C22×D7, C22×SD16, D4.D7, C2×C7⋊D4, C23×D7, C2×D4.D7, C22×C7⋊D4, C22×D4.D7

Smallest permutation representation of C22×D4.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 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)(169 183 176 190)(170 184 177 191)(171 185 178 192)(172 186 179 193)(173 187 180 194)(174 188 181 195)(175 189 182 196)(197 211 204 218)(198 212 205 219)(199 213 206 220)(200 214 207 221)(201 215 208 222)(202 216 209 223)(203 217 210 224)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(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)(127 211)(128 212)(129 213)(130 214)(131 215)(132 216)(133 217)(134 218)(135 219)(136 220)(137 221)(138 222)(139 223)(140 224)(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)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)
(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)
(1 180 8 173)(2 179 9 172)(3 178 10 171)(4 177 11 170)(5 176 12 169)(6 182 13 175)(7 181 14 174)(15 194 22 187)(16 193 23 186)(17 192 24 185)(18 191 25 184)(19 190 26 183)(20 196 27 189)(21 195 28 188)(29 208 36 201)(30 207 37 200)(31 206 38 199)(32 205 39 198)(33 204 40 197)(34 210 41 203)(35 209 42 202)(43 222 50 215)(44 221 51 214)(45 220 52 213)(46 219 53 212)(47 218 54 211)(48 224 55 217)(49 223 56 216)(57 124 64 117)(58 123 65 116)(59 122 66 115)(60 121 67 114)(61 120 68 113)(62 126 69 119)(63 125 70 118)(71 138 78 131)(72 137 79 130)(73 136 80 129)(74 135 81 128)(75 134 82 127)(76 140 83 133)(77 139 84 132)(85 152 92 145)(86 151 93 144)(87 150 94 143)(88 149 95 142)(89 148 96 141)(90 154 97 147)(91 153 98 146)(99 166 106 159)(100 165 107 158)(101 164 108 157)(102 163 109 156)(103 162 110 155)(104 168 111 161)(105 167 112 160)

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A ··· 8H 14A ··· 14U 14V ··· 14AS 28A ··· 28L order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 4 4 4 4 2 2 2 2 28 28 28 28 2 2 2 14 ··· 14 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 D4 D4 D7 SD16 D14 D14 C7⋊D4 C7⋊D4 D4.D7 kernel C22×D4.D7 C22×C7⋊C8 C2×D4.D7 C22×Dic14 D4×C2×C14 C2×C28 C22×C14 C22×D4 C2×C14 C22×C4 C2×D4 C2×C4 C23 C22 # reps 1 1 12 1 1 3 1 3 8 3 18 18 6 12

Matrix representation of C22×D4.D7 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 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 0 1 0 0 0 0 112 0
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 9 112 0 0 0 0 1 0 0 0 0 0 0 0 88 112 0 0 0 0 2 104 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 100 49 0 0 0 0 45 13 0 0 0 0 0 0 45 12 0 0 0 0 95 68 0 0 0 0 0 0 13 100 0 0 0 0 100 100

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,1,0,0,0,0,112,0,0,0,0,0,0,0,88,2,0,0,0,0,112,104,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[100,45,0,0,0,0,49,13,0,0,0,0,0,0,45,95,0,0,0,0,12,68,0,0,0,0,0,0,13,100,0,0,0,0,100,100] >;

C22×D4.D7 in GAP, Magma, Sage, TeX

C_2^2\times D_4.D_7
% in TeX

G:=Group("C2^2xD4.D7");
// GroupNames label

G:=SmallGroup(448,1247);
// by ID

G=gap.SmallGroup(448,1247);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,675,1684,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^2=e^7=1,f^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=f*c*f^-1=c^-1,c*e=e*c,d*e=e*d,f*d*f^-1=c*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽