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## G = C3×C23.63C23order 192 = 26·3

### Direct product of C3 and C23.63C23

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C23.63C23
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C6×C4⋊C4 — C3×C23.63C23
 Lower central C1 — C22 — C3×C23.63C23
 Upper central C1 — C22×C6 — C3×C23.63C23

Generators and relations for C3×C23.63C23
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=f2=d, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 226 in 154 conjugacy classes, 90 normal (62 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C22×C4, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C23.63C23, C3×C2.C42, C2×C4×C12, C6×C4⋊C4, C3×C23.63C23
Quotients:

Smallest permutation representation of C3×C23.63C23
Regular action on 192 points
Generators in S192
(1 109 105)(2 110 106)(3 111 107)(4 112 108)(5 144 140)(6 141 137)(7 142 138)(8 143 139)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(21 29 25)(22 30 26)(23 31 27)(24 32 28)(33 41 37)(34 42 38)(35 43 39)(36 44 40)(45 103 49)(46 104 50)(47 101 51)(48 102 52)(53 61 57)(54 62 58)(55 63 59)(56 64 60)(65 73 69)(66 74 70)(67 75 71)(68 76 72)(77 85 81)(78 86 82)(79 87 83)(80 88 84)(89 99 93)(90 100 94)(91 97 95)(92 98 96)(113 121 117)(114 122 118)(115 123 119)(116 124 120)(125 133 129)(126 134 130)(127 135 131)(128 136 132)(145 153 149)(146 154 150)(147 155 151)(148 156 152)(157 165 161)(158 166 162)(159 167 163)(160 168 164)(169 177 173)(170 178 174)(171 179 175)(172 180 176)(181 189 185)(182 190 186)(183 191 187)(184 192 188)
(1 9)(2 10)(3 11)(4 12)(5 102)(6 103)(7 104)(8 101)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 169)(78 170)(79 171)(80 172)(81 173)(82 174)(83 175)(84 176)(85 177)(86 178)(87 179)(88 180)(89 181)(90 182)(91 183)(92 184)(93 185)(94 186)(95 187)(96 188)(97 191)(98 192)(99 189)(100 190)
(1 147)(2 148)(3 145)(4 146)(5 190)(6 191)(7 192)(8 189)(9 55)(10 56)(11 53)(12 54)(13 59)(14 60)(15 57)(16 58)(17 63)(18 64)(19 61)(20 62)(21 67)(22 68)(23 65)(24 66)(25 71)(26 72)(27 69)(28 70)(29 75)(30 76)(31 73)(32 74)(33 79)(34 80)(35 77)(36 78)(37 83)(38 84)(39 81)(40 82)(41 87)(42 88)(43 85)(44 86)(45 91)(46 92)(47 89)(48 90)(49 95)(50 96)(51 93)(52 94)(97 103)(98 104)(99 101)(100 102)(105 151)(106 152)(107 149)(108 150)(109 155)(110 156)(111 153)(112 154)(113 159)(114 160)(115 157)(116 158)(117 163)(118 164)(119 161)(120 162)(121 167)(122 168)(123 165)(124 166)(125 171)(126 172)(127 169)(128 170)(129 175)(130 176)(131 173)(132 174)(133 179)(134 180)(135 177)(136 178)(137 183)(138 184)(139 181)(140 182)(141 187)(142 188)(143 185)(144 186)
(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 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)
(1 127 3 125)(2 36 4 34)(5 32 7 30)(6 121 8 123)(9 35 11 33)(10 128 12 126)(13 39 15 37)(14 132 16 130)(17 43 19 41)(18 136 20 134)(21 47 23 45)(22 140 24 138)(25 51 27 49)(26 144 28 142)(29 101 31 103)(38 106 40 108)(42 110 44 112)(46 114 48 116)(50 118 52 120)(53 79 55 77)(54 172 56 170)(57 83 59 81)(58 176 60 174)(61 87 63 85)(62 180 64 178)(65 91 67 89)(66 184 68 182)(69 95 71 93)(70 188 72 186)(73 97 75 99)(74 192 76 190)(78 146 80 148)(82 150 84 152)(86 154 88 156)(90 158 92 160)(94 162 96 164)(98 168 100 166)(102 124 104 122)(105 131 107 129)(109 135 111 133)(113 139 115 137)(117 143 119 141)(145 171 147 169)(149 175 151 173)(153 179 155 177)(157 183 159 181)(161 187 163 185)(165 191 167 189)
(1 21 9 113)(2 68 10 160)(3 23 11 115)(4 66 12 158)(5 178 102 86)(6 133 103 41)(7 180 104 88)(8 135 101 43)(13 117 105 25)(14 164 106 72)(15 119 107 27)(16 162 108 70)(17 121 109 29)(18 168 110 76)(19 123 111 31)(20 166 112 74)(22 56 114 148)(24 54 116 146)(26 60 118 152)(28 58 120 150)(30 64 122 156)(32 62 124 154)(33 137 125 45)(34 184 126 92)(35 139 127 47)(36 182 128 90)(37 141 129 49)(38 188 130 96)(39 143 131 51)(40 186 132 94)(42 192 134 98)(44 190 136 100)(46 80 138 172)(48 78 140 170)(50 84 142 176)(52 82 144 174)(53 157 145 65)(55 159 147 67)(57 161 149 69)(59 163 151 71)(61 165 153 73)(63 167 155 75)(77 181 169 89)(79 183 171 91)(81 185 173 93)(83 187 175 95)(85 189 177 99)(87 191 179 97)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4L 4M ··· 4T 6A ··· 6N 12A ··· 12X 12Y ··· 12AN order 1 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 Q8 C4○D4 C3×D4 C3×Q8 C3×C4○D4 kernel C3×C23.63C23 C3×C2.C42 C2×C4×C12 C6×C4⋊C4 C23.63C23 C3×C4⋊C4 C2.C42 C2×C42 C2×C4⋊C4 C4⋊C4 C2×C12 C2×C12 C2×C6 C2×C4 C2×C4 C22 # reps 1 4 1 2 2 8 8 2 4 16 2 2 8 4 4 16

Matrix representation of C3×C23.63C23 in GL5(𝔽13)

 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 6 0 0 0 2 0
,
 5 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 3 11 0 0 0 5 10
,
 12 0 0 0 0 0 8 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,6,0],[5,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,3,5,0,0,0,11,10],[12,0,0,0,0,0,8,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5] >;

C3×C23.63C23 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{63}C_2^3
% in TeX

G:=Group("C3xC2^3.63C2^3");
// GroupNames label

G:=SmallGroup(192,820);
// by ID

G=gap.SmallGroup(192,820);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,680,1094,142]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=f^2=d,g^2=b,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,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,d*f=f*d,d*g=g*d,f*g=g*f>;
// generators/relations

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