direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.83C23, (C2×C12).42Q8, (C2×C12).313D4, C22.76(C6×D4), C22.25(C6×Q8), C6.92(C22⋊Q8), C6.70(C4.4D4), C23.87(C22×C6), C6.30(C42.C2), C6.36(C42⋊2C2), C2.C42.13C6, (C22×C6).464C23, (C22×C12).405C22, C6.94(C22.D4), (C2×C4⋊C4).11C6, (C6×C4⋊C4).40C2, (C2×C4).5(C3×Q8), (C2×C4).20(C3×D4), (C2×C6).616(C2×D4), (C2×C6).113(C2×Q8), C2.8(C3×C4.4D4), C2.11(C3×C22⋊Q8), C2.5(C3×C42.C2), (C22×C4).14(C2×C6), C2.6(C3×C42⋊2C2), C22.43(C3×C4○D4), (C2×C6).224(C4○D4), (C3×C2.C42).6C2, C2.10(C3×C22.D4), SmallGroup(192,833)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.83C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=g2=b, f2=cb=bc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >
Subgroups: 218 in 134 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C4⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C23.83C23, C3×C2.C42, C3×C2.C42, C6×C4⋊C4, C3×C23.83C23
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C6×D4, C6×Q8, C3×C4○D4, C23.83C23, C3×C22⋊Q8, C3×C22.D4, C3×C4.4D4, C3×C42.C2, C3×C42⋊2C2, C3×C23.83C23
►Regular action on 192 points
Generators in S192
(1 157 57)(2 158 58)(3 159 59)(4 160 60)(5 96 101)(6 93 102)(7 94 103)(8 95 104)(9 17 109)(10 18 110)(11 19 111)(12 20 112)(13 105 113)(14 106 114)(15 107 115)(16 108 116)(21 29 121)(22 30 122)(23 31 123)(24 32 124)(25 117 125)(26 118 126)(27 119 127)(28 120 128)(33 41 133)(34 42 134)(35 43 135)(36 44 136)(37 129 137)(38 130 138)(39 131 139)(40 132 140)(45 53 145)(46 54 146)(47 55 147)(48 56 148)(49 141 149)(50 142 150)(51 143 151)(52 144 152)(61 153 161)(62 154 162)(63 155 163)(64 156 164)(65 73 165)(66 74 166)(67 75 167)(68 76 168)(69 77 169)(70 78 170)(71 79 171)(72 80 172)(81 173 181)(82 174 182)(83 175 183)(84 176 184)(85 177 185)(86 178 186)(87 179 187)(88 180 188)(89 98 189)(90 99 190)(91 100 191)(92 97 192)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)(97 99)(98 100)(101 103)(102 104)(105 107)(106 108)(109 111)(110 112)(113 115)(114 116)(117 119)(118 120)(121 123)(122 124)(125 127)(126 128)(129 131)(130 132)(133 135)(134 136)(137 139)(138 140)(141 143)(142 144)(145 147)(146 148)(149 151)(150 152)(153 155)(154 156)(157 159)(158 160)(161 163)(162 164)(165 167)(166 168)(169 171)(170 172)(173 175)(174 176)(177 179)(178 180)(181 183)(182 184)(185 187)(186 188)(189 191)(190 192)
(1 105)(2 106)(3 107)(4 108)(5 52)(6 49)(7 50)(8 51)(9 153)(10 154)(11 155)(12 156)(13 57)(14 58)(15 59)(16 60)(17 161)(18 162)(19 163)(20 164)(21 65)(22 66)(23 67)(24 68)(25 169)(26 170)(27 171)(28 172)(29 73)(30 74)(31 75)(32 76)(33 177)(34 178)(35 179)(36 180)(37 81)(38 82)(39 83)(40 84)(41 185)(42 186)(43 187)(44 188)(45 89)(46 90)(47 91)(48 92)(53 98)(54 99)(55 100)(56 97)(61 109)(62 110)(63 111)(64 112)(69 117)(70 118)(71 119)(72 120)(77 125)(78 126)(79 127)(80 128)(85 133)(86 134)(87 135)(88 136)(93 141)(94 142)(95 143)(96 144)(101 152)(102 149)(103 150)(104 151)(113 157)(114 158)(115 159)(116 160)(121 165)(122 166)(123 167)(124 168)(129 173)(130 174)(131 175)(132 176)(137 181)(138 182)(139 183)(140 184)(145 189)(146 190)(147 191)(148 192)
(1 11)(2 12)(3 9)(4 10)(5 146)(6 147)(7 148)(8 145)(13 63)(14 64)(15 61)(16 62)(17 159)(18 160)(19 157)(20 158)(21 71)(22 72)(23 69)(24 70)(25 167)(26 168)(27 165)(28 166)(29 79)(30 80)(31 77)(32 78)(33 175)(34 176)(35 173)(36 174)(37 87)(38 88)(39 85)(40 86)(41 183)(42 184)(43 181)(44 182)(45 95)(46 96)(47 93)(48 94)(49 191)(50 192)(51 189)(52 190)(53 104)(54 101)(55 102)(56 103)(57 111)(58 112)(59 109)(60 110)(65 119)(66 120)(67 117)(68 118)(73 127)(74 128)(75 125)(76 126)(81 135)(82 136)(83 133)(84 134)(89 143)(90 144)(91 141)(92 142)(97 150)(98 151)(99 152)(100 149)(105 155)(106 156)(107 153)(108 154)(113 163)(114 164)(115 161)(116 162)(121 171)(122 172)(123 169)(124 170)(129 179)(130 180)(131 177)(132 178)(137 187)(138 188)(139 185)(140 186)
(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)
(1 131 107 173)(2 130 108 176)(3 129 105 175)(4 132 106 174)(5 170 50 28)(6 169 51 27)(7 172 52 26)(8 171 49 25)(9 179 155 33)(10 178 156 36)(11 177 153 35)(12 180 154 34)(13 83 59 37)(14 82 60 40)(15 81 57 39)(16 84 58 38)(17 187 163 41)(18 186 164 44)(19 185 161 43)(20 188 162 42)(21 91 67 45)(22 90 68 48)(23 89 65 47)(24 92 66 46)(29 100 75 53)(30 99 76 56)(31 98 73 55)(32 97 74 54)(61 135 111 85)(62 134 112 88)(63 133 109 87)(64 136 110 86)(69 143 119 93)(70 142 120 96)(71 141 117 95)(72 144 118 94)(77 151 127 102)(78 150 128 101)(79 149 125 104)(80 152 126 103)(113 183 159 137)(114 182 160 140)(115 181 157 139)(116 184 158 138)(121 191 167 145)(122 190 168 148)(123 189 165 147)(124 192 166 146)
(1 67 3 65)(2 24 4 22)(5 82 7 84)(6 39 8 37)(9 119 11 117)(10 72 12 70)(13 123 15 121)(14 168 16 166)(17 127 19 125)(18 80 20 78)(21 105 23 107)(25 109 27 111)(26 62 28 64)(29 113 31 115)(30 158 32 160)(33 89 35 91)(34 46 36 48)(38 50 40 52)(41 98 43 100)(42 54 44 56)(45 179 47 177)(49 83 51 81)(53 187 55 185)(57 167 59 165)(58 124 60 122)(61 171 63 169)(66 106 68 108)(69 153 71 155)(73 157 75 159)(74 114 76 116)(77 161 79 163)(85 145 87 147)(86 190 88 192)(90 180 92 178)(93 131 95 129)(94 176 96 174)(97 186 99 188)(101 182 103 184)(102 139 104 137)(110 172 112 170)(118 154 120 156)(126 162 128 164)(130 142 132 144)(133 189 135 191)(134 146 136 148)(138 150 140 152)(141 175 143 173)(149 183 151 181)
G:=sub<Sym(192)| (1,157,57)(2,158,58)(3,159,59)(4,160,60)(5,96,101)(6,93,102)(7,94,103)(8,95,104)(9,17,109)(10,18,110)(11,19,111)(12,20,112)(13,105,113)(14,106,114)(15,107,115)(16,108,116)(21,29,121)(22,30,122)(23,31,123)(24,32,124)(25,117,125)(26,118,126)(27,119,127)(28,120,128)(33,41,133)(34,42,134)(35,43,135)(36,44,136)(37,129,137)(38,130,138)(39,131,139)(40,132,140)(45,53,145)(46,54,146)(47,55,147)(48,56,148)(49,141,149)(50,142,150)(51,143,151)(52,144,152)(61,153,161)(62,154,162)(63,155,163)(64,156,164)(65,73,165)(66,74,166)(67,75,167)(68,76,168)(69,77,169)(70,78,170)(71,79,171)(72,80,172)(81,173,181)(82,174,182)(83,175,183)(84,176,184)(85,177,185)(86,178,186)(87,179,187)(88,180,188)(89,98,189)(90,99,190)(91,100,191)(92,97,192), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192), (1,105)(2,106)(3,107)(4,108)(5,52)(6,49)(7,50)(8,51)(9,153)(10,154)(11,155)(12,156)(13,57)(14,58)(15,59)(16,60)(17,161)(18,162)(19,163)(20,164)(21,65)(22,66)(23,67)(24,68)(25,169)(26,170)(27,171)(28,172)(29,73)(30,74)(31,75)(32,76)(33,177)(34,178)(35,179)(36,180)(37,81)(38,82)(39,83)(40,84)(41,185)(42,186)(43,187)(44,188)(45,89)(46,90)(47,91)(48,92)(53,98)(54,99)(55,100)(56,97)(61,109)(62,110)(63,111)(64,112)(69,117)(70,118)(71,119)(72,120)(77,125)(78,126)(79,127)(80,128)(85,133)(86,134)(87,135)(88,136)(93,141)(94,142)(95,143)(96,144)(101,152)(102,149)(103,150)(104,151)(113,157)(114,158)(115,159)(116,160)(121,165)(122,166)(123,167)(124,168)(129,173)(130,174)(131,175)(132,176)(137,181)(138,182)(139,183)(140,184)(145,189)(146,190)(147,191)(148,192), (1,11)(2,12)(3,9)(4,10)(5,146)(6,147)(7,148)(8,145)(13,63)(14,64)(15,61)(16,62)(17,159)(18,160)(19,157)(20,158)(21,71)(22,72)(23,69)(24,70)(25,167)(26,168)(27,165)(28,166)(29,79)(30,80)(31,77)(32,78)(33,175)(34,176)(35,173)(36,174)(37,87)(38,88)(39,85)(40,86)(41,183)(42,184)(43,181)(44,182)(45,95)(46,96)(47,93)(48,94)(49,191)(50,192)(51,189)(52,190)(53,104)(54,101)(55,102)(56,103)(57,111)(58,112)(59,109)(60,110)(65,119)(66,120)(67,117)(68,118)(73,127)(74,128)(75,125)(76,126)(81,135)(82,136)(83,133)(84,134)(89,143)(90,144)(91,141)(92,142)(97,150)(98,151)(99,152)(100,149)(105,155)(106,156)(107,153)(108,154)(113,163)(114,164)(115,161)(116,162)(121,171)(122,172)(123,169)(124,170)(129,179)(130,180)(131,177)(132,178)(137,187)(138,188)(139,185)(140,186), (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), (1,131,107,173)(2,130,108,176)(3,129,105,175)(4,132,106,174)(5,170,50,28)(6,169,51,27)(7,172,52,26)(8,171,49,25)(9,179,155,33)(10,178,156,36)(11,177,153,35)(12,180,154,34)(13,83,59,37)(14,82,60,40)(15,81,57,39)(16,84,58,38)(17,187,163,41)(18,186,164,44)(19,185,161,43)(20,188,162,42)(21,91,67,45)(22,90,68,48)(23,89,65,47)(24,92,66,46)(29,100,75,53)(30,99,76,56)(31,98,73,55)(32,97,74,54)(61,135,111,85)(62,134,112,88)(63,133,109,87)(64,136,110,86)(69,143,119,93)(70,142,120,96)(71,141,117,95)(72,144,118,94)(77,151,127,102)(78,150,128,101)(79,149,125,104)(80,152,126,103)(113,183,159,137)(114,182,160,140)(115,181,157,139)(116,184,158,138)(121,191,167,145)(122,190,168,148)(123,189,165,147)(124,192,166,146), (1,67,3,65)(2,24,4,22)(5,82,7,84)(6,39,8,37)(9,119,11,117)(10,72,12,70)(13,123,15,121)(14,168,16,166)(17,127,19,125)(18,80,20,78)(21,105,23,107)(25,109,27,111)(26,62,28,64)(29,113,31,115)(30,158,32,160)(33,89,35,91)(34,46,36,48)(38,50,40,52)(41,98,43,100)(42,54,44,56)(45,179,47,177)(49,83,51,81)(53,187,55,185)(57,167,59,165)(58,124,60,122)(61,171,63,169)(66,106,68,108)(69,153,71,155)(73,157,75,159)(74,114,76,116)(77,161,79,163)(85,145,87,147)(86,190,88,192)(90,180,92,178)(93,131,95,129)(94,176,96,174)(97,186,99,188)(101,182,103,184)(102,139,104,137)(110,172,112,170)(118,154,120,156)(126,162,128,164)(130,142,132,144)(133,189,135,191)(134,146,136,148)(138,150,140,152)(141,175,143,173)(149,183,151,181)>;
G:=Group( (1,157,57)(2,158,58)(3,159,59)(4,160,60)(5,96,101)(6,93,102)(7,94,103)(8,95,104)(9,17,109)(10,18,110)(11,19,111)(12,20,112)(13,105,113)(14,106,114)(15,107,115)(16,108,116)(21,29,121)(22,30,122)(23,31,123)(24,32,124)(25,117,125)(26,118,126)(27,119,127)(28,120,128)(33,41,133)(34,42,134)(35,43,135)(36,44,136)(37,129,137)(38,130,138)(39,131,139)(40,132,140)(45,53,145)(46,54,146)(47,55,147)(48,56,148)(49,141,149)(50,142,150)(51,143,151)(52,144,152)(61,153,161)(62,154,162)(63,155,163)(64,156,164)(65,73,165)(66,74,166)(67,75,167)(68,76,168)(69,77,169)(70,78,170)(71,79,171)(72,80,172)(81,173,181)(82,174,182)(83,175,183)(84,176,184)(85,177,185)(86,178,186)(87,179,187)(88,180,188)(89,98,189)(90,99,190)(91,100,191)(92,97,192), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192), (1,105)(2,106)(3,107)(4,108)(5,52)(6,49)(7,50)(8,51)(9,153)(10,154)(11,155)(12,156)(13,57)(14,58)(15,59)(16,60)(17,161)(18,162)(19,163)(20,164)(21,65)(22,66)(23,67)(24,68)(25,169)(26,170)(27,171)(28,172)(29,73)(30,74)(31,75)(32,76)(33,177)(34,178)(35,179)(36,180)(37,81)(38,82)(39,83)(40,84)(41,185)(42,186)(43,187)(44,188)(45,89)(46,90)(47,91)(48,92)(53,98)(54,99)(55,100)(56,97)(61,109)(62,110)(63,111)(64,112)(69,117)(70,118)(71,119)(72,120)(77,125)(78,126)(79,127)(80,128)(85,133)(86,134)(87,135)(88,136)(93,141)(94,142)(95,143)(96,144)(101,152)(102,149)(103,150)(104,151)(113,157)(114,158)(115,159)(116,160)(121,165)(122,166)(123,167)(124,168)(129,173)(130,174)(131,175)(132,176)(137,181)(138,182)(139,183)(140,184)(145,189)(146,190)(147,191)(148,192), (1,11)(2,12)(3,9)(4,10)(5,146)(6,147)(7,148)(8,145)(13,63)(14,64)(15,61)(16,62)(17,159)(18,160)(19,157)(20,158)(21,71)(22,72)(23,69)(24,70)(25,167)(26,168)(27,165)(28,166)(29,79)(30,80)(31,77)(32,78)(33,175)(34,176)(35,173)(36,174)(37,87)(38,88)(39,85)(40,86)(41,183)(42,184)(43,181)(44,182)(45,95)(46,96)(47,93)(48,94)(49,191)(50,192)(51,189)(52,190)(53,104)(54,101)(55,102)(56,103)(57,111)(58,112)(59,109)(60,110)(65,119)(66,120)(67,117)(68,118)(73,127)(74,128)(75,125)(76,126)(81,135)(82,136)(83,133)(84,134)(89,143)(90,144)(91,141)(92,142)(97,150)(98,151)(99,152)(100,149)(105,155)(106,156)(107,153)(108,154)(113,163)(114,164)(115,161)(116,162)(121,171)(122,172)(123,169)(124,170)(129,179)(130,180)(131,177)(132,178)(137,187)(138,188)(139,185)(140,186), (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), (1,131,107,173)(2,130,108,176)(3,129,105,175)(4,132,106,174)(5,170,50,28)(6,169,51,27)(7,172,52,26)(8,171,49,25)(9,179,155,33)(10,178,156,36)(11,177,153,35)(12,180,154,34)(13,83,59,37)(14,82,60,40)(15,81,57,39)(16,84,58,38)(17,187,163,41)(18,186,164,44)(19,185,161,43)(20,188,162,42)(21,91,67,45)(22,90,68,48)(23,89,65,47)(24,92,66,46)(29,100,75,53)(30,99,76,56)(31,98,73,55)(32,97,74,54)(61,135,111,85)(62,134,112,88)(63,133,109,87)(64,136,110,86)(69,143,119,93)(70,142,120,96)(71,141,117,95)(72,144,118,94)(77,151,127,102)(78,150,128,101)(79,149,125,104)(80,152,126,103)(113,183,159,137)(114,182,160,140)(115,181,157,139)(116,184,158,138)(121,191,167,145)(122,190,168,148)(123,189,165,147)(124,192,166,146), (1,67,3,65)(2,24,4,22)(5,82,7,84)(6,39,8,37)(9,119,11,117)(10,72,12,70)(13,123,15,121)(14,168,16,166)(17,127,19,125)(18,80,20,78)(21,105,23,107)(25,109,27,111)(26,62,28,64)(29,113,31,115)(30,158,32,160)(33,89,35,91)(34,46,36,48)(38,50,40,52)(41,98,43,100)(42,54,44,56)(45,179,47,177)(49,83,51,81)(53,187,55,185)(57,167,59,165)(58,124,60,122)(61,171,63,169)(66,106,68,108)(69,153,71,155)(73,157,75,159)(74,114,76,116)(77,161,79,163)(85,145,87,147)(86,190,88,192)(90,180,92,178)(93,131,95,129)(94,176,96,174)(97,186,99,188)(101,182,103,184)(102,139,104,137)(110,172,112,170)(118,154,120,156)(126,162,128,164)(130,142,132,144)(133,189,135,191)(134,146,136,148)(138,150,140,152)(141,175,143,173)(149,183,151,181) );
G=PermutationGroup([[(1,157,57),(2,158,58),(3,159,59),(4,160,60),(5,96,101),(6,93,102),(7,94,103),(8,95,104),(9,17,109),(10,18,110),(11,19,111),(12,20,112),(13,105,113),(14,106,114),(15,107,115),(16,108,116),(21,29,121),(22,30,122),(23,31,123),(24,32,124),(25,117,125),(26,118,126),(27,119,127),(28,120,128),(33,41,133),(34,42,134),(35,43,135),(36,44,136),(37,129,137),(38,130,138),(39,131,139),(40,132,140),(45,53,145),(46,54,146),(47,55,147),(48,56,148),(49,141,149),(50,142,150),(51,143,151),(52,144,152),(61,153,161),(62,154,162),(63,155,163),(64,156,164),(65,73,165),(66,74,166),(67,75,167),(68,76,168),(69,77,169),(70,78,170),(71,79,171),(72,80,172),(81,173,181),(82,174,182),(83,175,183),(84,176,184),(85,177,185),(86,178,186),(87,179,187),(88,180,188),(89,98,189),(90,99,190),(91,100,191),(92,97,192)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(70,72),(73,75),(74,76),(77,79),(78,80),(81,83),(82,84),(85,87),(86,88),(89,91),(90,92),(93,95),(94,96),(97,99),(98,100),(101,103),(102,104),(105,107),(106,108),(109,111),(110,112),(113,115),(114,116),(117,119),(118,120),(121,123),(122,124),(125,127),(126,128),(129,131),(130,132),(133,135),(134,136),(137,139),(138,140),(141,143),(142,144),(145,147),(146,148),(149,151),(150,152),(153,155),(154,156),(157,159),(158,160),(161,163),(162,164),(165,167),(166,168),(169,171),(170,172),(173,175),(174,176),(177,179),(178,180),(181,183),(182,184),(185,187),(186,188),(189,191),(190,192)], [(1,105),(2,106),(3,107),(4,108),(5,52),(6,49),(7,50),(8,51),(9,153),(10,154),(11,155),(12,156),(13,57),(14,58),(15,59),(16,60),(17,161),(18,162),(19,163),(20,164),(21,65),(22,66),(23,67),(24,68),(25,169),(26,170),(27,171),(28,172),(29,73),(30,74),(31,75),(32,76),(33,177),(34,178),(35,179),(36,180),(37,81),(38,82),(39,83),(40,84),(41,185),(42,186),(43,187),(44,188),(45,89),(46,90),(47,91),(48,92),(53,98),(54,99),(55,100),(56,97),(61,109),(62,110),(63,111),(64,112),(69,117),(70,118),(71,119),(72,120),(77,125),(78,126),(79,127),(80,128),(85,133),(86,134),(87,135),(88,136),(93,141),(94,142),(95,143),(96,144),(101,152),(102,149),(103,150),(104,151),(113,157),(114,158),(115,159),(116,160),(121,165),(122,166),(123,167),(124,168),(129,173),(130,174),(131,175),(132,176),(137,181),(138,182),(139,183),(140,184),(145,189),(146,190),(147,191),(148,192)], [(1,11),(2,12),(3,9),(4,10),(5,146),(6,147),(7,148),(8,145),(13,63),(14,64),(15,61),(16,62),(17,159),(18,160),(19,157),(20,158),(21,71),(22,72),(23,69),(24,70),(25,167),(26,168),(27,165),(28,166),(29,79),(30,80),(31,77),(32,78),(33,175),(34,176),(35,173),(36,174),(37,87),(38,88),(39,85),(40,86),(41,183),(42,184),(43,181),(44,182),(45,95),(46,96),(47,93),(48,94),(49,191),(50,192),(51,189),(52,190),(53,104),(54,101),(55,102),(56,103),(57,111),(58,112),(59,109),(60,110),(65,119),(66,120),(67,117),(68,118),(73,127),(74,128),(75,125),(76,126),(81,135),(82,136),(83,133),(84,134),(89,143),(90,144),(91,141),(92,142),(97,150),(98,151),(99,152),(100,149),(105,155),(106,156),(107,153),(108,154),(113,163),(114,164),(115,161),(116,162),(121,171),(122,172),(123,169),(124,170),(129,179),(130,180),(131,177),(132,178),(137,187),(138,188),(139,185),(140,186)], [(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)], [(1,131,107,173),(2,130,108,176),(3,129,105,175),(4,132,106,174),(5,170,50,28),(6,169,51,27),(7,172,52,26),(8,171,49,25),(9,179,155,33),(10,178,156,36),(11,177,153,35),(12,180,154,34),(13,83,59,37),(14,82,60,40),(15,81,57,39),(16,84,58,38),(17,187,163,41),(18,186,164,44),(19,185,161,43),(20,188,162,42),(21,91,67,45),(22,90,68,48),(23,89,65,47),(24,92,66,46),(29,100,75,53),(30,99,76,56),(31,98,73,55),(32,97,74,54),(61,135,111,85),(62,134,112,88),(63,133,109,87),(64,136,110,86),(69,143,119,93),(70,142,120,96),(71,141,117,95),(72,144,118,94),(77,151,127,102),(78,150,128,101),(79,149,125,104),(80,152,126,103),(113,183,159,137),(114,182,160,140),(115,181,157,139),(116,184,158,138),(121,191,167,145),(122,190,168,148),(123,189,165,147),(124,192,166,146)], [(1,67,3,65),(2,24,4,22),(5,82,7,84),(6,39,8,37),(9,119,11,117),(10,72,12,70),(13,123,15,121),(14,168,16,166),(17,127,19,125),(18,80,20,78),(21,105,23,107),(25,109,27,111),(26,62,28,64),(29,113,31,115),(30,158,32,160),(33,89,35,91),(34,46,36,48),(38,50,40,52),(41,98,43,100),(42,54,44,56),(45,179,47,177),(49,83,51,81),(53,187,55,185),(57,167,59,165),(58,124,60,122),(61,171,63,169),(66,106,68,108),(69,153,71,155),(73,157,75,159),(74,114,76,116),(77,161,79,163),(85,145,87,147),(86,190,88,192),(90,180,92,178),(93,131,95,129),(94,176,96,174),(97,186,99,188),(101,182,103,184),(102,139,104,137),(110,172,112,170),(118,154,120,156),(126,162,128,164),(130,142,132,144),(133,189,135,191),(134,146,136,148),(138,150,140,152),(141,175,143,173),(149,183,151,181)]])
66 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4N | 6A | ··· | 6N | 12A | ··· | 12AB |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | Q8 | C4○D4 | C3×D4 | C3×Q8 | C3×C4○D4 |
| kernel | C3×C23.83C23 | C3×C2.C42 | C6×C4⋊C4 | C23.83C23 | C2.C42 | C2×C4⋊C4 | C2×C12 | C2×C12 | C2×C6 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 5 | 2 | 2 | 10 | 4 | 2 | 2 | 10 | 4 | 4 | 20 |
Matrix representation of C3×C23.83C23 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 4 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 12 | 0 | 0 |
| 0 | 0 | 11 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 9 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 4 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 8 | 11 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[5,4,0,0,0,0,7,8,0,0,0,0,0,0,7,11,0,0,0,0,12,6,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[12,7,0,0,0,0,9,1,0,0,0,0,0,0,6,4,0,0,0,0,1,7,0,0,0,0,0,0,8,0,0,0,0,0,0,5],[8,12,0,0,0,0,11,5,0,0,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C3×C23.83C23 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{83}C_2^3 % in TeX
G:=Group("C3xC2^3.83C2^3"); // GroupNames label
G:=SmallGroup(192,833);
// by ID
G=gap.SmallGroup(192,833);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,512,1094,1059,142]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c*b=b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
// generators/relations