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G = C3×C23.84C23  order 192 = 26·3

Direct product of C3 and C23.84C23

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

Aliases: C3×C23.84C23, C2.C42.7C6, C23.88(C22×C6), C6.37(C42⋊2C2), (C22×C12).37C22, (C22×C6).465C23, (C22×C4).15(C2×C6), C2.7(C3×C42⋊2C2), C22.44(C3×C4○D4), (C2×C6).225(C4○D4), (C3×C2.C42).7C2, SmallGroup(192,834)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C3×C23.84C23
Chief series
C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C3×C2.C42 — C3×C23.84C23
Lower central
C1 — C23 — C3×C23.84C23
Upper central
C1 — C22×C6 — C3×C23.84C23

Generators and relations for C3×C23.84C23
 G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=bcd, f2=cb=bc, g2=b, 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: 202 in 118 conjugacy classes, 62 normal (6 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C2×C12, C22×C6, C2.C42, C22×C12, C23.84C23, C3×C2.C42, C3×C23.84C23
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C22×C6, C42⋊2C2, C3×C4○D4, C23.84C23, C3×C42⋊2C2, C3×C23.84C23

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

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

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

(1 108)(2 105)(3 106)(4 107)(5 53)(6 54)(7 55)(8 56)(9 150)(10 151)(11 152)(12 149)(13 154)(14 155)(15 156)(16 153)(17 160)(18 157)(19 158)(20 159)(21 162)(22 163)(23 164)(24 161)(25 168)(26 165)(27 166)(28 167)(29 170)(30 171)(31 172)(32 169)(33 176)(34 173)(35 174)(36 175)(37 178)(38 179)(39 180)(40 177)(41 184)(42 181)(43 182)(44 183)(45 186)(46 187)(47 188)(48 185)(49 192)(50 189)(51 190)(52 191)(57 104)(58 101)(59 102)(60 103)(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)(97 145)(98 146)(99 147)(100 148)

(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)

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

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

66 conjugacy classes

class 1 2A···2G3A3B4A···4N6A···6N12A···12AB
order12···2334···46···612···12
size11···1114···41···14···4

66 irreducible representations

dim111122
type++
imageC1C2C3C6C4○D4C3×C4○D4
kernelC3×C23.84C23C3×C2.C42C23.84C23C2.C42C2×C6C22
# reps172141428

Matrix representation of C3×C23.84C23 ►in GL6(𝔽13)

900000
090000
001000
000100
000030
000003
,
100000
010000
0012000
0001200
000010
000001
,
1200000
0120000
001000
000100
0000120
0000012
,
100000
010000
0012000
0001200
0000120
0000012
,
370000
6100000
000800
005000
000029
0000411
,
500000
050000
003900
0091000
0000106
000073
,
010000
100000
000100
0012000
000001
000010

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,2,4,0,0,0,0,9,11],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,3,9,0,0,0,0,9,10,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C3×C23.84C23 in GAP, Magma, Sage, TeX 

C_3\times C_2^3._{84}C_2^3
 
% in TeX
 
G:=Group("C3xC2^3.84C2^3");
 
// GroupNames label
 
G:=SmallGroup(192,834);
 
// by ID
 
G=gap.SmallGroup(192,834);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,1176,365,512,1094,1059,142]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=b*c*d,f^2=c*b=b*c,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*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
 

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