direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22.58C24, C6.1252- 1+4, C42.56(C2×C6), C42.C2.7C6, (C2×C6).384C24, (C4×C12).297C22, (C2×C12).685C23, C22.58(C23×C6), C2.17(C3×2- 1+4), C4⋊C4.36(C2×C6), (C2×C4).44(C22×C6), (C3×C4⋊C4).253C22, (C3×C42.C2).14C2, SmallGroup(192,1453)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.58C24
G = < a,b,c,d,e,f,g | a3=b2=c2=1, d2=g2=b, e2=f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg-1=bcd, fef-1=bce, fg=gf >
Subgroups: 202 in 172 conjugacy classes, 142 normal (6 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, C4⋊C4, C2×C12, C42.C2, C4×C12, C3×C4⋊C4, C22.58C24, C3×C42.C2, C3×C22.58C24
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2- 1+4, C23×C6, C22.58C24, C3×2- 1+4, C3×C22.58C24
►Regular action on 192 points
Generators in S192
(1 11 103)(2 12 104)(3 9 101)(4 10 102)(5 49 141)(6 50 142)(7 51 143)(8 52 144)(13 105 109)(14 106 110)(15 107 111)(16 108 112)(17 21 113)(18 22 114)(19 23 115)(20 24 116)(25 117 121)(26 118 122)(27 119 123)(28 120 124)(29 33 125)(30 34 126)(31 35 127)(32 36 128)(37 129 133)(38 130 134)(39 131 135)(40 132 136)(41 45 137)(42 46 138)(43 47 139)(44 48 140)(53 145 149)(54 146 150)(55 147 151)(56 148 152)(57 61 153)(58 62 154)(59 63 155)(60 64 156)(65 157 161)(66 158 162)(67 159 163)(68 160 164)(69 73 165)(70 74 166)(71 75 167)(72 76 168)(77 169 173)(78 170 174)(79 171 175)(80 172 176)(81 85 177)(82 86 178)(83 87 179)(84 88 180)(89 181 185)(90 182 186)(91 183 187)(92 184 188)(93 100 189)(94 97 190)(95 98 191)(96 99 192)
(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)
(1 147)(2 148)(3 145)(4 146)(5 100)(6 97)(7 98)(8 99)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 163)(24 164)(25 165)(26 166)(27 167)(28 168)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)
(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)
(1 171 147 31)(2 170 148 30)(3 169 145 29)(4 172 146 32)(5 121 100 73)(6 124 97 76)(7 123 98 75)(8 122 99 74)(9 173 149 33)(10 176 150 36)(11 175 151 35)(12 174 152 34)(13 177 153 37)(14 180 154 40)(15 179 155 39)(16 178 156 38)(17 181 157 41)(18 184 158 44)(19 183 159 43)(20 182 160 42)(21 185 161 45)(22 188 162 48)(23 187 163 47)(24 186 164 46)(25 189 165 49)(26 192 166 52)(27 191 167 51)(28 190 168 50)(53 125 101 77)(54 128 102 80)(55 127 103 79)(56 126 104 78)(57 129 105 81)(58 132 106 84)(59 131 107 83)(60 130 108 82)(61 133 109 85)(62 136 110 88)(63 135 111 87)(64 134 112 86)(65 137 113 89)(66 140 114 92)(67 139 115 91)(68 138 116 90)(69 141 117 93)(70 144 118 96)(71 143 119 95)(72 142 120 94)
(1 159 147 19)(2 20 148 160)(3 157 145 17)(4 18 146 158)(5 87 100 135)(6 136 97 88)(7 85 98 133)(8 134 99 86)(9 161 149 21)(10 22 150 162)(11 163 151 23)(12 24 152 164)(13 165 153 25)(14 26 154 166)(15 167 155 27)(16 28 156 168)(29 43 169 183)(30 184 170 44)(31 41 171 181)(32 182 172 42)(33 47 173 187)(34 188 174 48)(35 45 175 185)(36 186 176 46)(37 51 177 191)(38 192 178 52)(39 49 179 189)(40 190 180 50)(53 113 101 65)(54 66 102 114)(55 115 103 67)(56 68 104 116)(57 117 105 69)(58 70 106 118)(59 119 107 71)(60 72 108 120)(61 121 109 73)(62 74 110 122)(63 123 111 75)(64 76 112 124)(77 91 125 139)(78 140 126 92)(79 89 127 137)(80 138 128 90)(81 95 129 143)(82 144 130 96)(83 93 131 141)(84 142 132 94)
(1 105 3 107)(2 60 4 58)(5 47 7 45)(6 186 8 188)(9 111 11 109)(10 62 12 64)(13 101 15 103)(14 56 16 54)(17 119 19 117)(18 70 20 72)(21 123 23 121)(22 74 24 76)(25 113 27 115)(26 68 28 66)(29 129 31 131)(30 84 32 82)(33 133 35 135)(34 88 36 86)(37 127 39 125)(38 78 40 80)(41 141 43 143)(42 96 44 94)(46 99 48 97)(49 139 51 137)(50 90 52 92)(53 155 55 153)(57 145 59 147)(61 149 63 151)(65 167 67 165)(69 157 71 159)(73 161 75 163)(77 177 79 179)(81 171 83 169)(85 175 87 173)(89 189 91 191)(93 183 95 181)(98 185 100 187)(102 154 104 156)(106 148 108 146)(110 152 112 150)(114 166 116 168)(118 160 120 158)(122 164 124 162)(126 180 128 178)(130 170 132 172)(134 174 136 176)(138 192 140 190)(142 182 144 184)
G:=sub<Sym(192)| (1,11,103)(2,12,104)(3,9,101)(4,10,102)(5,49,141)(6,50,142)(7,51,143)(8,52,144)(13,105,109)(14,106,110)(15,107,111)(16,108,112)(17,21,113)(18,22,114)(19,23,115)(20,24,116)(25,117,121)(26,118,122)(27,119,123)(28,120,124)(29,33,125)(30,34,126)(31,35,127)(32,36,128)(37,129,133)(38,130,134)(39,131,135)(40,132,136)(41,45,137)(42,46,138)(43,47,139)(44,48,140)(53,145,149)(54,146,150)(55,147,151)(56,148,152)(57,61,153)(58,62,154)(59,63,155)(60,64,156)(65,157,161)(66,158,162)(67,159,163)(68,160,164)(69,73,165)(70,74,166)(71,75,167)(72,76,168)(77,169,173)(78,170,174)(79,171,175)(80,172,176)(81,85,177)(82,86,178)(83,87,179)(84,88,180)(89,181,185)(90,182,186)(91,183,187)(92,184,188)(93,100,189)(94,97,190)(95,98,191)(96,99,192), (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), (1,147)(2,148)(3,145)(4,146)(5,100)(6,97)(7,98)(8,99)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,166)(27,167)(28,168)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144), (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), (1,171,147,31)(2,170,148,30)(3,169,145,29)(4,172,146,32)(5,121,100,73)(6,124,97,76)(7,123,98,75)(8,122,99,74)(9,173,149,33)(10,176,150,36)(11,175,151,35)(12,174,152,34)(13,177,153,37)(14,180,154,40)(15,179,155,39)(16,178,156,38)(17,181,157,41)(18,184,158,44)(19,183,159,43)(20,182,160,42)(21,185,161,45)(22,188,162,48)(23,187,163,47)(24,186,164,46)(25,189,165,49)(26,192,166,52)(27,191,167,51)(28,190,168,50)(53,125,101,77)(54,128,102,80)(55,127,103,79)(56,126,104,78)(57,129,105,81)(58,132,106,84)(59,131,107,83)(60,130,108,82)(61,133,109,85)(62,136,110,88)(63,135,111,87)(64,134,112,86)(65,137,113,89)(66,140,114,92)(67,139,115,91)(68,138,116,90)(69,141,117,93)(70,144,118,96)(71,143,119,95)(72,142,120,94), (1,159,147,19)(2,20,148,160)(3,157,145,17)(4,18,146,158)(5,87,100,135)(6,136,97,88)(7,85,98,133)(8,134,99,86)(9,161,149,21)(10,22,150,162)(11,163,151,23)(12,24,152,164)(13,165,153,25)(14,26,154,166)(15,167,155,27)(16,28,156,168)(29,43,169,183)(30,184,170,44)(31,41,171,181)(32,182,172,42)(33,47,173,187)(34,188,174,48)(35,45,175,185)(36,186,176,46)(37,51,177,191)(38,192,178,52)(39,49,179,189)(40,190,180,50)(53,113,101,65)(54,66,102,114)(55,115,103,67)(56,68,104,116)(57,117,105,69)(58,70,106,118)(59,119,107,71)(60,72,108,120)(61,121,109,73)(62,74,110,122)(63,123,111,75)(64,76,112,124)(77,91,125,139)(78,140,126,92)(79,89,127,137)(80,138,128,90)(81,95,129,143)(82,144,130,96)(83,93,131,141)(84,142,132,94), (1,105,3,107)(2,60,4,58)(5,47,7,45)(6,186,8,188)(9,111,11,109)(10,62,12,64)(13,101,15,103)(14,56,16,54)(17,119,19,117)(18,70,20,72)(21,123,23,121)(22,74,24,76)(25,113,27,115)(26,68,28,66)(29,129,31,131)(30,84,32,82)(33,133,35,135)(34,88,36,86)(37,127,39,125)(38,78,40,80)(41,141,43,143)(42,96,44,94)(46,99,48,97)(49,139,51,137)(50,90,52,92)(53,155,55,153)(57,145,59,147)(61,149,63,151)(65,167,67,165)(69,157,71,159)(73,161,75,163)(77,177,79,179)(81,171,83,169)(85,175,87,173)(89,189,91,191)(93,183,95,181)(98,185,100,187)(102,154,104,156)(106,148,108,146)(110,152,112,150)(114,166,116,168)(118,160,120,158)(122,164,124,162)(126,180,128,178)(130,170,132,172)(134,174,136,176)(138,192,140,190)(142,182,144,184)>;
G:=Group( (1,11,103)(2,12,104)(3,9,101)(4,10,102)(5,49,141)(6,50,142)(7,51,143)(8,52,144)(13,105,109)(14,106,110)(15,107,111)(16,108,112)(17,21,113)(18,22,114)(19,23,115)(20,24,116)(25,117,121)(26,118,122)(27,119,123)(28,120,124)(29,33,125)(30,34,126)(31,35,127)(32,36,128)(37,129,133)(38,130,134)(39,131,135)(40,132,136)(41,45,137)(42,46,138)(43,47,139)(44,48,140)(53,145,149)(54,146,150)(55,147,151)(56,148,152)(57,61,153)(58,62,154)(59,63,155)(60,64,156)(65,157,161)(66,158,162)(67,159,163)(68,160,164)(69,73,165)(70,74,166)(71,75,167)(72,76,168)(77,169,173)(78,170,174)(79,171,175)(80,172,176)(81,85,177)(82,86,178)(83,87,179)(84,88,180)(89,181,185)(90,182,186)(91,183,187)(92,184,188)(93,100,189)(94,97,190)(95,98,191)(96,99,192), (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), (1,147)(2,148)(3,145)(4,146)(5,100)(6,97)(7,98)(8,99)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,166)(27,167)(28,168)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144), (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), (1,171,147,31)(2,170,148,30)(3,169,145,29)(4,172,146,32)(5,121,100,73)(6,124,97,76)(7,123,98,75)(8,122,99,74)(9,173,149,33)(10,176,150,36)(11,175,151,35)(12,174,152,34)(13,177,153,37)(14,180,154,40)(15,179,155,39)(16,178,156,38)(17,181,157,41)(18,184,158,44)(19,183,159,43)(20,182,160,42)(21,185,161,45)(22,188,162,48)(23,187,163,47)(24,186,164,46)(25,189,165,49)(26,192,166,52)(27,191,167,51)(28,190,168,50)(53,125,101,77)(54,128,102,80)(55,127,103,79)(56,126,104,78)(57,129,105,81)(58,132,106,84)(59,131,107,83)(60,130,108,82)(61,133,109,85)(62,136,110,88)(63,135,111,87)(64,134,112,86)(65,137,113,89)(66,140,114,92)(67,139,115,91)(68,138,116,90)(69,141,117,93)(70,144,118,96)(71,143,119,95)(72,142,120,94), (1,159,147,19)(2,20,148,160)(3,157,145,17)(4,18,146,158)(5,87,100,135)(6,136,97,88)(7,85,98,133)(8,134,99,86)(9,161,149,21)(10,22,150,162)(11,163,151,23)(12,24,152,164)(13,165,153,25)(14,26,154,166)(15,167,155,27)(16,28,156,168)(29,43,169,183)(30,184,170,44)(31,41,171,181)(32,182,172,42)(33,47,173,187)(34,188,174,48)(35,45,175,185)(36,186,176,46)(37,51,177,191)(38,192,178,52)(39,49,179,189)(40,190,180,50)(53,113,101,65)(54,66,102,114)(55,115,103,67)(56,68,104,116)(57,117,105,69)(58,70,106,118)(59,119,107,71)(60,72,108,120)(61,121,109,73)(62,74,110,122)(63,123,111,75)(64,76,112,124)(77,91,125,139)(78,140,126,92)(79,89,127,137)(80,138,128,90)(81,95,129,143)(82,144,130,96)(83,93,131,141)(84,142,132,94), (1,105,3,107)(2,60,4,58)(5,47,7,45)(6,186,8,188)(9,111,11,109)(10,62,12,64)(13,101,15,103)(14,56,16,54)(17,119,19,117)(18,70,20,72)(21,123,23,121)(22,74,24,76)(25,113,27,115)(26,68,28,66)(29,129,31,131)(30,84,32,82)(33,133,35,135)(34,88,36,86)(37,127,39,125)(38,78,40,80)(41,141,43,143)(42,96,44,94)(46,99,48,97)(49,139,51,137)(50,90,52,92)(53,155,55,153)(57,145,59,147)(61,149,63,151)(65,167,67,165)(69,157,71,159)(73,161,75,163)(77,177,79,179)(81,171,83,169)(85,175,87,173)(89,189,91,191)(93,183,95,181)(98,185,100,187)(102,154,104,156)(106,148,108,146)(110,152,112,150)(114,166,116,168)(118,160,120,158)(122,164,124,162)(126,180,128,178)(130,170,132,172)(134,174,136,176)(138,192,140,190)(142,182,144,184) );
G=PermutationGroup([[(1,11,103),(2,12,104),(3,9,101),(4,10,102),(5,49,141),(6,50,142),(7,51,143),(8,52,144),(13,105,109),(14,106,110),(15,107,111),(16,108,112),(17,21,113),(18,22,114),(19,23,115),(20,24,116),(25,117,121),(26,118,122),(27,119,123),(28,120,124),(29,33,125),(30,34,126),(31,35,127),(32,36,128),(37,129,133),(38,130,134),(39,131,135),(40,132,136),(41,45,137),(42,46,138),(43,47,139),(44,48,140),(53,145,149),(54,146,150),(55,147,151),(56,148,152),(57,61,153),(58,62,154),(59,63,155),(60,64,156),(65,157,161),(66,158,162),(67,159,163),(68,160,164),(69,73,165),(70,74,166),(71,75,167),(72,76,168),(77,169,173),(78,170,174),(79,171,175),(80,172,176),(81,85,177),(82,86,178),(83,87,179),(84,88,180),(89,181,185),(90,182,186),(91,183,187),(92,184,188),(93,100,189),(94,97,190),(95,98,191),(96,99,192)], [(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)], [(1,147),(2,148),(3,145),(4,146),(5,100),(6,97),(7,98),(8,99),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,161),(22,162),(23,163),(24,164),(25,165),(26,166),(27,167),(28,168),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(51,191),(52,192),(53,101),(54,102),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144)], [(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)], [(1,171,147,31),(2,170,148,30),(3,169,145,29),(4,172,146,32),(5,121,100,73),(6,124,97,76),(7,123,98,75),(8,122,99,74),(9,173,149,33),(10,176,150,36),(11,175,151,35),(12,174,152,34),(13,177,153,37),(14,180,154,40),(15,179,155,39),(16,178,156,38),(17,181,157,41),(18,184,158,44),(19,183,159,43),(20,182,160,42),(21,185,161,45),(22,188,162,48),(23,187,163,47),(24,186,164,46),(25,189,165,49),(26,192,166,52),(27,191,167,51),(28,190,168,50),(53,125,101,77),(54,128,102,80),(55,127,103,79),(56,126,104,78),(57,129,105,81),(58,132,106,84),(59,131,107,83),(60,130,108,82),(61,133,109,85),(62,136,110,88),(63,135,111,87),(64,134,112,86),(65,137,113,89),(66,140,114,92),(67,139,115,91),(68,138,116,90),(69,141,117,93),(70,144,118,96),(71,143,119,95),(72,142,120,94)], [(1,159,147,19),(2,20,148,160),(3,157,145,17),(4,18,146,158),(5,87,100,135),(6,136,97,88),(7,85,98,133),(8,134,99,86),(9,161,149,21),(10,22,150,162),(11,163,151,23),(12,24,152,164),(13,165,153,25),(14,26,154,166),(15,167,155,27),(16,28,156,168),(29,43,169,183),(30,184,170,44),(31,41,171,181),(32,182,172,42),(33,47,173,187),(34,188,174,48),(35,45,175,185),(36,186,176,46),(37,51,177,191),(38,192,178,52),(39,49,179,189),(40,190,180,50),(53,113,101,65),(54,66,102,114),(55,115,103,67),(56,68,104,116),(57,117,105,69),(58,70,106,118),(59,119,107,71),(60,72,108,120),(61,121,109,73),(62,74,110,122),(63,123,111,75),(64,76,112,124),(77,91,125,139),(78,140,126,92),(79,89,127,137),(80,138,128,90),(81,95,129,143),(82,144,130,96),(83,93,131,141),(84,142,132,94)], [(1,105,3,107),(2,60,4,58),(5,47,7,45),(6,186,8,188),(9,111,11,109),(10,62,12,64),(13,101,15,103),(14,56,16,54),(17,119,19,117),(18,70,20,72),(21,123,23,121),(22,74,24,76),(25,113,27,115),(26,68,28,66),(29,129,31,131),(30,84,32,82),(33,133,35,135),(34,88,36,86),(37,127,39,125),(38,78,40,80),(41,141,43,143),(42,96,44,94),(46,99,48,97),(49,139,51,137),(50,90,52,92),(53,155,55,153),(57,145,59,147),(61,149,63,151),(65,167,67,165),(69,157,71,159),(73,161,75,163),(77,177,79,179),(81,171,83,169),(85,175,87,173),(89,189,91,191),(93,183,95,181),(98,185,100,187),(102,154,104,156),(106,148,108,146),(110,152,112,150),(114,166,116,168),(118,160,120,158),(122,164,124,162),(126,180,128,178),(130,170,132,172),(134,174,136,176),(138,192,140,190),(142,182,144,184)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4O | 6A | ··· | 6F | 12A | ··· | 12AD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | - | |||
| image | C1 | C2 | C3 | C6 | 2- 1+4 | C3×2- 1+4 |
| kernel | C3×C22.58C24 | C3×C42.C2 | C22.58C24 | C42.C2 | C6 | C2 |
| # reps | 1 | 15 | 2 | 30 | 3 | 6 |
Matrix representation of C3×C22.58C24 ►in GL8(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 9 | 2 | 1 | 2 | 0 | 0 | 0 | 0 |
| 2 | 4 | 2 | 12 | 0 | 0 | 0 | 0 |
| 1 | 2 | 9 | 2 | 0 | 0 | 0 | 0 |
| 2 | 12 | 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 | 4 | 8 |
| 0 | 0 | 0 | 0 | 2 | 1 | 5 | 9 |
| 0 | 0 | 0 | 0 | 4 | 8 | 1 | 2 |
| 0 | 0 | 0 | 0 | 5 | 9 | 11 | 12 |
| 5 | 10 | 11 | 9 | 0 | 0 | 0 | 0 |
| 10 | 8 | 9 | 2 | 0 | 0 | 0 | 0 |
| 2 | 4 | 8 | 3 | 0 | 0 | 0 | 0 |
| 4 | 11 | 3 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 6 | 5 | 8 |
| 0 | 0 | 0 | 0 | 6 | 2 | 8 | 5 |
| 0 | 0 | 0 | 0 | 5 | 8 | 11 | 7 |
| 0 | 0 | 0 | 0 | 8 | 5 | 7 | 11 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[9,2,1,2,0,0,0,0,2,4,2,12,0,0,0,0,1,2,9,2,0,0,0,0,2,12,2,4,0,0,0,0,0,0,0,0,12,2,4,5,0,0,0,0,11,1,8,9,0,0,0,0,4,5,1,11,0,0,0,0,8,9,2,12],[5,10,2,4,0,0,0,0,10,8,4,11,0,0,0,0,11,9,8,3,0,0,0,0,9,2,3,5,0,0,0,0,0,0,0,0,2,6,5,8,0,0,0,0,6,2,8,5,0,0,0,0,5,8,11,7,0,0,0,0,8,5,7,11],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C3×C22.58C24 in GAP, Magma, Sage, TeX
C_3\times C_2^2._{58}C_2^4 % in TeX
G:=Group("C3xC2^2.58C2^4"); // GroupNames label
G:=SmallGroup(192,1453);
// by ID
G=gap.SmallGroup(192,1453);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,680,2102,1563,520,4259,794,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=1,d^2=g^2=b,e^2=f^2=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*c=c*b,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
// generators/relations