direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.32D6, C24.85D6, C23.64D12, C23.19Dic6, (C22×C12)⋊9C4, (C23×C4).8S3, C6⋊(C2.C42), (C2×C6).27C42, C23.69(C4×S3), (C23×C12).1C2, C6.27(C2×C42), (C22×C4)⋊7Dic3, (C22×C6).24Q8, (C22×Dic3)⋊9C4, C22.59(C2×D12), (C22×C6).190D4, (C22×C4).420D6, C22.53(D6⋊C4), (C23×C6).95C22, (C23×Dic3).6C2, C22.22(C4×Dic3), C23.47(C2×Dic3), C22.30(C2×Dic6), C23.106(C3⋊D4), (C22×C6).359C23, C23.309(C22×S3), C22.23(C4⋊Dic3), C22.27(Dic3⋊C4), (C22×C12).480C22, C22.25(C22×Dic3), C22.32(C6.D4), (C22×Dic3).194C22, C6.54(C2×C4⋊C4), C2.3(C2×D6⋊C4), (C2×C12)⋊34(C2×C4), (C2×C4)⋊9(C2×Dic3), C22.64(S3×C2×C4), C2.3(C2×C4⋊Dic3), (C2×C6).42(C2×Q8), C2.15(C2×C4×Dic3), (C2×C6).55(C4⋊C4), (C2×C6).545(C2×D4), C6.64(C2×C22⋊C4), C2.3(C2×Dic3⋊C4), C3⋊2(C2×C2.C42), (C2×Dic3)⋊20(C2×C4), (C22×C6).97(C2×C4), C2.2(C2×C6.D4), C22.83(C2×C3⋊D4), (C2×C6).69(C22⋊C4), (C2×C6).140(C22×C4), SmallGroup(192,767)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.32D6
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, C23.32D6 [×4], C23×Dic3 [×2], C23×C12, C2×C23.32D6
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, C23.32D6 [×8], C2×C4×Dic3, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×D6⋊C4 [×2], C2×C6.D4, C2×C23.32D6
►Regular action on 192 points
Generators in S192
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 37)(9 38)(10 39)(11 40)(12 41)(13 147)(14 148)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 155)(22 156)(23 145)(24 146)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(49 84)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 121)(85 179)(86 180)(87 169)(88 170)(89 171)(90 172)(91 173)(92 174)(93 175)(94 176)(95 177)(96 178)(109 144)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(157 186)(158 187)(159 188)(160 189)(161 190)(162 191)(163 192)(164 181)(165 182)(166 183)(167 184)(168 185)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 61)(10 62)(11 63)(12 64)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 166)(26 167)(27 168)(28 157)(29 158)(30 159)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 145)(83 146)(84 147)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 109)(95 110)(96 111)(97 183)(98 184)(99 185)(100 186)(101 187)(102 188)(103 189)(104 190)(105 191)(106 192)(107 181)(108 182)(133 177)(134 178)(135 179)(136 180)(137 169)(138 170)(139 171)(140 172)(141 173)(142 174)(143 175)(144 176)
(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)
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 169)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 97)(24 98)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 95)(38 96)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 185)(50 186)(51 187)(52 188)(53 189)(54 190)(55 191)(56 192)(57 181)(58 182)(59 183)(60 184)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 133)(73 157)(74 158)(75 159)(76 160)(77 161)(78 162)(79 163)(80 164)(81 165)(82 166)(83 167)(84 168)(109 132)(110 121)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 131)
(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)
(1 150 144 165)(2 35 133 74)(3 148 134 163)(4 33 135 84)(5 146 136 161)(6 31 137 82)(7 156 138 159)(8 29 139 80)(9 154 140 157)(10 27 141 78)(11 152 142 167)(12 25 143 76)(13 129 191 85)(14 111 192 44)(15 127 181 95)(16 109 182 42)(17 125 183 93)(18 119 184 40)(19 123 185 91)(20 117 186 38)(21 121 187 89)(22 115 188 48)(23 131 189 87)(24 113 190 46)(26 63 77 174)(28 61 79 172)(30 71 81 170)(32 69 83 180)(34 67 73 178)(36 65 75 176)(37 101 116 57)(39 99 118 55)(41 97 120 53)(43 107 110 51)(45 105 112 49)(47 103 114 59)(50 96 106 128)(52 94 108 126)(54 92 98 124)(56 90 100 122)(58 88 102 132)(60 86 104 130)(62 168 173 153)(64 166 175 151)(66 164 177 149)(68 162 179 147)(70 160 169 145)(72 158 171 155)
G:=sub<Sym(192)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,147)(14,148)(15,149)(16,150)(17,151)(18,152)(19,153)(20,154)(21,155)(22,156)(23,145)(24,146)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(49,84)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,121)(85,179)(86,180)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(109,144)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,181)(165,182)(166,183)(167,184)(168,185), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,61)(10,62)(11,63)(12,64)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,166)(26,167)(27,168)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,145)(83,146)(84,147)(85,112)(86,113)(87,114)(88,115)(89,116)(90,117)(91,118)(92,119)(93,120)(94,109)(95,110)(96,111)(97,183)(98,184)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,181)(108,182)(133,177)(134,178)(135,179)(136,180)(137,169)(138,170)(139,171)(140,172)(141,173)(142,174)(143,175)(144,176), (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), (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,169)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,97)(24,98)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,95)(38,96)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,185)(50,186)(51,187)(52,188)(53,189)(54,190)(55,191)(56,192)(57,181)(58,182)(59,183)(60,184)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,133)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168)(109,132)(110,121)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131), (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), (1,150,144,165)(2,35,133,74)(3,148,134,163)(4,33,135,84)(5,146,136,161)(6,31,137,82)(7,156,138,159)(8,29,139,80)(9,154,140,157)(10,27,141,78)(11,152,142,167)(12,25,143,76)(13,129,191,85)(14,111,192,44)(15,127,181,95)(16,109,182,42)(17,125,183,93)(18,119,184,40)(19,123,185,91)(20,117,186,38)(21,121,187,89)(22,115,188,48)(23,131,189,87)(24,113,190,46)(26,63,77,174)(28,61,79,172)(30,71,81,170)(32,69,83,180)(34,67,73,178)(36,65,75,176)(37,101,116,57)(39,99,118,55)(41,97,120,53)(43,107,110,51)(45,105,112,49)(47,103,114,59)(50,96,106,128)(52,94,108,126)(54,92,98,124)(56,90,100,122)(58,88,102,132)(60,86,104,130)(62,168,173,153)(64,166,175,151)(66,164,177,149)(68,162,179,147)(70,160,169,145)(72,158,171,155)>;
G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,147)(14,148)(15,149)(16,150)(17,151)(18,152)(19,153)(20,154)(21,155)(22,156)(23,145)(24,146)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(49,84)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,121)(85,179)(86,180)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(109,144)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,181)(165,182)(166,183)(167,184)(168,185), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,61)(10,62)(11,63)(12,64)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,166)(26,167)(27,168)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,145)(83,146)(84,147)(85,112)(86,113)(87,114)(88,115)(89,116)(90,117)(91,118)(92,119)(93,120)(94,109)(95,110)(96,111)(97,183)(98,184)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,181)(108,182)(133,177)(134,178)(135,179)(136,180)(137,169)(138,170)(139,171)(140,172)(141,173)(142,174)(143,175)(144,176), (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), (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,169)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,97)(24,98)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,95)(38,96)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,185)(50,186)(51,187)(52,188)(53,189)(54,190)(55,191)(56,192)(57,181)(58,182)(59,183)(60,184)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,133)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168)(109,132)(110,121)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131), (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), (1,150,144,165)(2,35,133,74)(3,148,134,163)(4,33,135,84)(5,146,136,161)(6,31,137,82)(7,156,138,159)(8,29,139,80)(9,154,140,157)(10,27,141,78)(11,152,142,167)(12,25,143,76)(13,129,191,85)(14,111,192,44)(15,127,181,95)(16,109,182,42)(17,125,183,93)(18,119,184,40)(19,123,185,91)(20,117,186,38)(21,121,187,89)(22,115,188,48)(23,131,189,87)(24,113,190,46)(26,63,77,174)(28,61,79,172)(30,71,81,170)(32,69,83,180)(34,67,73,178)(36,65,75,176)(37,101,116,57)(39,99,118,55)(41,97,120,53)(43,107,110,51)(45,105,112,49)(47,103,114,59)(50,96,106,128)(52,94,108,126)(54,92,98,124)(56,90,100,122)(58,88,102,132)(60,86,104,130)(62,168,173,153)(64,166,175,151)(66,164,177,149)(68,162,179,147)(70,160,169,145)(72,158,171,155) );
G=PermutationGroup([(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,37),(9,38),(10,39),(11,40),(12,41),(13,147),(14,148),(15,149),(16,150),(17,151),(18,152),(19,153),(20,154),(21,155),(22,156),(23,145),(24,146),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(49,84),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,80),(58,81),(59,82),(60,83),(61,122),(62,123),(63,124),(64,125),(65,126),(66,127),(67,128),(68,129),(69,130),(70,131),(71,132),(72,121),(85,179),(86,180),(87,169),(88,170),(89,171),(90,172),(91,173),(92,174),(93,175),(94,176),(95,177),(96,178),(109,144),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138),(116,139),(117,140),(118,141),(119,142),(120,143),(157,186),(158,187),(159,188),(160,189),(161,190),(162,191),(163,192),(164,181),(165,182),(166,183),(167,184),(168,185)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,61),(10,62),(11,63),(12,64),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,166),(26,167),(27,168),(28,157),(29,158),(30,159),(31,160),(32,161),(33,162),(34,163),(35,164),(36,165),(37,121),(38,122),(39,123),(40,124),(41,125),(42,126),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(73,148),(74,149),(75,150),(76,151),(77,152),(78,153),(79,154),(80,155),(81,156),(82,145),(83,146),(84,147),(85,112),(86,113),(87,114),(88,115),(89,116),(90,117),(91,118),(92,119),(93,120),(94,109),(95,110),(96,111),(97,183),(98,184),(99,185),(100,186),(101,187),(102,188),(103,189),(104,190),(105,191),(106,192),(107,181),(108,182),(133,177),(134,178),(135,179),(136,180),(137,169),(138,170),(139,171),(140,172),(141,173),(142,174),(143,175),(144,176)], [(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)], [(1,170),(2,171),(3,172),(4,173),(5,174),(6,175),(7,176),(8,177),(9,178),(10,179),(11,180),(12,169),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,97),(24,98),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,95),(38,96),(39,85),(40,86),(41,87),(42,88),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(49,185),(50,186),(51,187),(52,188),(53,189),(54,190),(55,191),(56,192),(57,181),(58,182),(59,183),(60,184),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,141),(69,142),(70,143),(71,144),(72,133),(73,157),(74,158),(75,159),(76,160),(77,161),(78,162),(79,163),(80,164),(81,165),(82,166),(83,167),(84,168),(109,132),(110,121),(111,122),(112,123),(113,124),(114,125),(115,126),(116,127),(117,128),(118,129),(119,130),(120,131)], [(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)], [(1,150,144,165),(2,35,133,74),(3,148,134,163),(4,33,135,84),(5,146,136,161),(6,31,137,82),(7,156,138,159),(8,29,139,80),(9,154,140,157),(10,27,141,78),(11,152,142,167),(12,25,143,76),(13,129,191,85),(14,111,192,44),(15,127,181,95),(16,109,182,42),(17,125,183,93),(18,119,184,40),(19,123,185,91),(20,117,186,38),(21,121,187,89),(22,115,188,48),(23,131,189,87),(24,113,190,46),(26,63,77,174),(28,61,79,172),(30,71,81,170),(32,69,83,180),(34,67,73,178),(36,65,75,176),(37,101,116,57),(39,99,118,55),(41,97,120,53),(43,107,110,51),(45,105,112,49),(47,103,114,59),(50,96,106,128),(52,94,108,126),(54,92,98,124),(56,90,100,122),(58,88,102,132),(60,86,104,130),(62,168,173,153),(64,166,175,151),(66,164,177,149),(68,162,179,147),(70,160,169,145),(72,158,171,155)])
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×C23.32D6 | C23.32D6 | 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×C23.32D6 ►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×C23.32D6 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{32}D_6 % in TeX
G:=Group("C2xC2^3.32D6"); // 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