Copied to
clipboard

## G = C2×C6.C42order 192 = 26·3

### Direct product of C2 and C6.C42

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C6.C42
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C23×Dic3 — C2×C6.C42
 Lower central C3 — C6 — C2×C6.C42
 Upper central C1 — C24 — C23×C4

Generators and relations for C2×C6.C42
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 664 in 330 conjugacy classes, 183 normal (21 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×12], C22 [×3], C22 [×32], C6 [×3], C6 [×12], C2×C4 [×4], C2×C4 [×44], C23, C23 [×14], Dic3 [×8], C12 [×4], C2×C6 [×3], C2×C6 [×32], C22×C4 [×6], C22×C4 [×24], C24, C2×Dic3 [×8], C2×Dic3 [×24], C2×C12 [×4], C2×C12 [×12], C22×C6, C22×C6 [×14], C2.C42 [×4], C23×C4, C23×C4 [×2], C22×Dic3 [×12], C22×Dic3 [×8], C22×C12 [×6], C22×C12 [×4], C23×C6, C2×C2.C42, C6.C42 [×4], C23×Dic3 [×2], C23×C12, C2×C6.C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3, C2×C4 [×18], D4 [×6], Q8 [×2], C23, Dic3 [×4], D6 [×3], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×4], D6⋊C4 [×8], C6.D4 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C22×Dic3, C2×C3⋊D4 [×2], C2×C2.C42, C6.C42 [×8], C2×C4×Dic3, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×D6⋊C4 [×2], C2×C6.D4, C2×C6.C42

Smallest permutation representation of C2×C6.C42
Regular action on 192 points
Generators in S192
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 157)(10 158)(11 159)(12 160)(13 131)(14 132)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 129)(24 130)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 146)(38 147)(39 148)(40 149)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 145)(61 120)(62 109)(63 110)(64 111)(65 112)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 175)(86 176)(87 177)(88 178)(89 179)(90 180)(91 169)(92 170)(93 171)(94 172)(95 173)(96 174)(133 189)(134 190)(135 191)(136 192)(137 181)(138 182)(139 183)(140 184)(141 185)(142 186)(143 187)(144 188)
(1 176)(2 177)(3 178)(4 179)(5 180)(6 169)(7 170)(8 171)(9 172)(10 173)(11 174)(12 175)(13 148)(14 149)(15 150)(16 151)(17 152)(18 153)(19 154)(20 155)(21 156)(22 145)(23 146)(24 147)(25 81)(26 82)(27 83)(28 84)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 129)(38 130)(39 131)(40 132)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 105)(50 106)(51 107)(52 108)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 191)(62 192)(63 181)(64 182)(65 183)(66 184)(67 185)(68 186)(69 187)(70 188)(71 189)(72 190)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 157)(95 158)(96 159)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(118 133)(119 134)(120 135)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 141)(10 142)(11 143)(12 144)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 97)(21 98)(22 99)(23 100)(24 101)(25 48)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 85)(71 86)(72 87)(73 132)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(81 128)(82 129)(83 130)(84 131)(109 179)(110 180)(111 169)(112 170)(113 171)(114 172)(115 173)(116 174)(117 175)(118 176)(119 177)(120 178)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 181)(166 182)(167 183)(168 184)
(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)
(1 13 112 52)(2 107 113 147)(3 23 114 50)(4 105 115 145)(5 21 116 60)(6 103 117 155)(7 19 118 58)(8 101 119 153)(9 17 120 56)(10 99 109 151)(11 15 110 54)(12 97 111 149)(14 175 53 138)(16 173 55 136)(18 171 57 134)(20 169 59 144)(22 179 49 142)(24 177 51 140)(25 192 122 95)(26 157 123 61)(27 190 124 93)(28 167 125 71)(29 188 126 91)(30 165 127 69)(31 186 128 89)(32 163 129 67)(33 184 130 87)(34 161 131 65)(35 182 132 85)(36 159 121 63)(37 185 76 88)(38 162 77 66)(39 183 78 86)(40 160 79 64)(41 181 80 96)(42 158 81 62)(43 191 82 94)(44 168 83 72)(45 189 84 92)(46 166 73 70)(47 187 74 90)(48 164 75 68)(98 174 150 137)(100 172 152 135)(102 170 154 133)(104 180 156 143)(106 178 146 141)(108 176 148 139)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2O 3 4A ··· 4H 4I ··· 4X 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + - - + + - + image C1 C2 C2 C2 C4 C4 S3 D4 Q8 Dic3 D6 D6 Dic6 C4×S3 D12 C3⋊D4 kernel C2×C6.C42 C6.C42 C23×Dic3 C23×C12 C22×Dic3 C22×C12 C23×C4 C22×C6 C22×C6 C22×C4 C22×C4 C24 C23 C23 C23 C23 # reps 1 4 2 1 16 8 1 6 2 4 2 1 4 8 4 8

Matrix representation of C2×C6.C42 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 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 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 11 0 0 0 0 0 12 0 0 0 0 0 0 5 8 0 0 0 0 5 0
,
 8 0 0 0 0 0 0 1 0 0 0 0 0 0 10 7 0 0 0 0 10 3 0 0 0 0 0 0 7 3 0 0 0 0 10 6

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,5,5,0,0,0,0,8,0],[8,0,0,0,0,0,0,1,0,0,0,0,0,0,10,10,0,0,0,0,7,3,0,0,0,0,0,0,7,10,0,0,0,0,3,6] >;

C2×C6.C42 in GAP, Magma, Sage, TeX

C_2\times C_6.C_4^2
% in TeX

G:=Group("C2xC6.C4^2");
// GroupNames label

G:=SmallGroup(192,767);
// by ID

G=gap.SmallGroup(192,767);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,100,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=b*c*d,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,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^5>;
// generators/relations

׿
×
𝔽