direct product, metabelian, supersoluble, monomial
Aliases: C2×C33⋊15D4, C63⋊4C2, C62⋊34D6, C33⋊37(C2×D4), (C2×C62)⋊12S3, (C32×C6)⋊15D4, C6⋊3(C32⋊7D4), (C3×C62)⋊16C22, C33⋊5C4⋊13C22, C23⋊2(C33⋊C2), (C32×C6).104C23, C3⋊4(C2×C32⋊7D4), (C3×C6)⋊13(C3⋊D4), (C22×C6)⋊4(C3⋊S3), C6.48(C22×C3⋊S3), C32⋊22(C2×C3⋊D4), (C2×C33⋊5C4)⋊10C2, C22⋊3(C2×C33⋊C2), (C3×C6).193(C22×S3), (C22×C33⋊C2)⋊6C2, (C2×C33⋊C2)⋊14C22, C2.10(C22×C33⋊C2), (C2×C6)⋊11(C2×C3⋊S3), SmallGroup(432,729)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C33⋊15D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=b-1, cd=dc, ece-1=fcf=c-1, ede-1=fdf=d-1, fef=e-1 >
Subgroups: 3944 in 756 conjugacy classes, 235 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, D6, C2×C6, C2×C6, C2×D4, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C33, C3⋊Dic3, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C33⋊C2, C32×C6, C32×C6, C32×C6, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C2×C62, C33⋊5C4, C2×C33⋊C2, C2×C33⋊C2, C3×C62, C3×C62, C3×C62, C2×C32⋊7D4, C2×C33⋊5C4, C33⋊15D4, C22×C33⋊C2, C63, C2×C33⋊15D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C2×C3⋊S3, C2×C3⋊D4, C33⋊C2, C32⋊7D4, C22×C3⋊S3, C2×C33⋊C2, C2×C32⋊7D4, C33⋊15D4, C22×C33⋊C2, C2×C33⋊15D4
►On 216 points
Generators in S216
(1 10)(2 11)(3 12)(4 9)(5 83)(6 84)(7 81)(8 82)(13 141)(14 142)(15 143)(16 144)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 106)(26 107)(27 108)(28 105)(29 139)(30 140)(31 137)(32 138)(33 115)(34 116)(35 113)(36 114)(37 119)(38 120)(39 117)(40 118)(41 123)(42 124)(43 121)(44 122)(45 127)(46 128)(47 125)(48 126)(49 131)(50 132)(51 129)(52 130)(53 154)(54 155)(55 156)(56 153)(57 172)(58 169)(59 170)(60 171)(61 176)(62 173)(63 174)(64 175)(65 180)(66 177)(67 178)(68 179)(69 184)(70 181)(71 182)(72 183)(73 188)(74 185)(75 186)(76 187)(77 215)(78 216)(79 213)(80 214)(85 196)(86 193)(87 194)(88 195)(89 200)(90 197)(91 198)(92 199)(93 204)(94 201)(95 202)(96 203)(97 208)(98 205)(99 206)(100 207)(101 212)(102 209)(103 210)(104 211)(109 167)(110 168)(111 165)(112 166)(133 164)(134 161)(135 162)(136 163)(157 192)(158 189)(159 190)(160 191)
(1 162 182)(2 183 163)(3 164 184)(4 181 161)(5 177 132)(6 129 178)(7 179 130)(8 131 180)(9 70 134)(10 135 71)(11 72 136)(12 133 69)(13 74 30)(14 31 75)(15 76 32)(16 29 73)(17 216 87)(18 88 213)(19 214 85)(20 86 215)(21 109 89)(22 90 110)(23 111 91)(24 92 112)(25 45 63)(26 64 46)(27 47 61)(28 62 48)(33 156 96)(34 93 153)(35 154 94)(36 95 155)(37 158 98)(38 99 159)(39 160 100)(40 97 157)(41 170 104)(42 101 171)(43 172 102)(44 103 169)(49 65 82)(50 83 66)(51 67 84)(52 81 68)(53 201 113)(54 114 202)(55 203 115)(56 116 204)(57 209 121)(58 122 210)(59 211 123)(60 124 212)(77 148 193)(78 194 145)(79 146 195)(80 196 147)(105 173 126)(106 127 174)(107 175 128)(108 125 176)(117 191 207)(118 208 192)(119 189 205)(120 206 190)(137 186 142)(138 143 187)(139 188 144)(140 141 185)(149 167 200)(150 197 168)(151 165 198)(152 199 166)
(1 211 112)(2 109 212)(3 209 110)(4 111 210)(5 78 99)(6 100 79)(7 80 97)(8 98 77)(9 165 103)(10 104 166)(11 167 101)(12 102 168)(13 153 128)(14 125 154)(15 155 126)(16 127 156)(17 120 50)(18 51 117)(19 118 52)(20 49 119)(21 124 163)(22 164 121)(23 122 161)(24 162 123)(25 115 188)(26 185 116)(27 113 186)(28 187 114)(29 174 96)(30 93 175)(31 176 94)(32 95 173)(33 73 106)(34 107 74)(35 75 108)(36 105 76)(37 148 131)(38 132 145)(39 146 129)(40 130 147)(41 152 135)(42 136 149)(43 150 133)(44 134 151)(45 55 144)(46 141 56)(47 53 142)(48 143 54)(57 90 184)(58 181 91)(59 92 182)(60 183 89)(61 201 137)(62 138 202)(63 203 139)(64 140 204)(65 189 86)(66 87 190)(67 191 88)(68 85 192)(69 172 197)(70 198 169)(71 170 199)(72 200 171)(81 214 208)(82 205 215)(83 216 206)(84 207 213)(157 179 196)(158 193 180)(159 177 194)(160 195 178)
(1 49 201)(2 202 50)(3 51 203)(4 204 52)(5 72 155)(6 156 69)(7 70 153)(8 154 71)(9 93 130)(10 131 94)(11 95 132)(12 129 96)(13 97 169)(14 170 98)(15 99 171)(16 172 100)(17 109 62)(18 63 110)(19 111 64)(20 61 112)(21 28 87)(22 88 25)(23 26 85)(24 86 27)(29 102 39)(30 40 103)(31 104 37)(32 38 101)(33 133 178)(34 179 134)(35 135 180)(36 177 136)(41 158 75)(42 76 159)(43 160 73)(44 74 157)(45 90 213)(46 214 91)(47 92 215)(48 216 89)(53 182 82)(54 83 183)(55 184 84)(56 81 181)(57 207 144)(58 141 208)(59 205 142)(60 143 206)(65 113 162)(66 163 114)(67 115 164)(68 161 116)(77 125 199)(78 200 126)(79 127 197)(80 198 128)(105 194 149)(106 150 195)(107 196 151)(108 152 193)(117 139 209)(118 210 140)(119 137 211)(120 212 138)(121 191 188)(122 185 192)(123 189 186)(124 187 190)(145 167 173)(146 174 168)(147 165 175)(148 176 166)
(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)(193 194 195 196)(197 198 199 200)(201 202 203 204)(205 206 207 208)(209 210 211 212)(213 214 215 216)
(1 9)(2 12)(3 11)(4 10)(5 84)(6 83)(7 82)(8 81)(13 142)(14 141)(15 144)(16 143)(17 146)(18 145)(19 148)(20 147)(21 150)(22 149)(23 152)(24 151)(25 105)(26 108)(27 107)(28 106)(29 138)(30 137)(31 140)(32 139)(33 114)(34 113)(35 116)(36 115)(37 118)(38 117)(39 120)(40 119)(41 122)(42 121)(43 124)(44 123)(45 126)(46 125)(47 128)(48 127)(49 130)(50 129)(51 132)(52 131)(53 153)(54 156)(55 155)(56 154)(57 171)(58 170)(59 169)(60 172)(61 175)(62 174)(63 173)(64 176)(65 179)(66 178)(67 177)(68 180)(69 183)(70 182)(71 181)(72 184)(73 187)(74 186)(75 185)(76 188)(77 214)(78 213)(79 216)(80 215)(85 193)(86 196)(87 195)(88 194)(89 197)(90 200)(91 199)(92 198)(93 201)(94 204)(95 203)(96 202)(97 205)(98 208)(99 207)(100 206)(101 209)(102 212)(103 211)(104 210)(109 168)(110 167)(111 166)(112 165)(133 163)(134 162)(135 161)(136 164)(157 189)(158 192)(159 191)(160 190)
G:=sub<Sym(216)| (1,10)(2,11)(3,12)(4,9)(5,83)(6,84)(7,81)(8,82)(13,141)(14,142)(15,143)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,106)(26,107)(27,108)(28,105)(29,139)(30,140)(31,137)(32,138)(33,115)(34,116)(35,113)(36,114)(37,119)(38,120)(39,117)(40,118)(41,123)(42,124)(43,121)(44,122)(45,127)(46,128)(47,125)(48,126)(49,131)(50,132)(51,129)(52,130)(53,154)(54,155)(55,156)(56,153)(57,172)(58,169)(59,170)(60,171)(61,176)(62,173)(63,174)(64,175)(65,180)(66,177)(67,178)(68,179)(69,184)(70,181)(71,182)(72,183)(73,188)(74,185)(75,186)(76,187)(77,215)(78,216)(79,213)(80,214)(85,196)(86,193)(87,194)(88,195)(89,200)(90,197)(91,198)(92,199)(93,204)(94,201)(95,202)(96,203)(97,208)(98,205)(99,206)(100,207)(101,212)(102,209)(103,210)(104,211)(109,167)(110,168)(111,165)(112,166)(133,164)(134,161)(135,162)(136,163)(157,192)(158,189)(159,190)(160,191), (1,162,182)(2,183,163)(3,164,184)(4,181,161)(5,177,132)(6,129,178)(7,179,130)(8,131,180)(9,70,134)(10,135,71)(11,72,136)(12,133,69)(13,74,30)(14,31,75)(15,76,32)(16,29,73)(17,216,87)(18,88,213)(19,214,85)(20,86,215)(21,109,89)(22,90,110)(23,111,91)(24,92,112)(25,45,63)(26,64,46)(27,47,61)(28,62,48)(33,156,96)(34,93,153)(35,154,94)(36,95,155)(37,158,98)(38,99,159)(39,160,100)(40,97,157)(41,170,104)(42,101,171)(43,172,102)(44,103,169)(49,65,82)(50,83,66)(51,67,84)(52,81,68)(53,201,113)(54,114,202)(55,203,115)(56,116,204)(57,209,121)(58,122,210)(59,211,123)(60,124,212)(77,148,193)(78,194,145)(79,146,195)(80,196,147)(105,173,126)(106,127,174)(107,175,128)(108,125,176)(117,191,207)(118,208,192)(119,189,205)(120,206,190)(137,186,142)(138,143,187)(139,188,144)(140,141,185)(149,167,200)(150,197,168)(151,165,198)(152,199,166), (1,211,112)(2,109,212)(3,209,110)(4,111,210)(5,78,99)(6,100,79)(7,80,97)(8,98,77)(9,165,103)(10,104,166)(11,167,101)(12,102,168)(13,153,128)(14,125,154)(15,155,126)(16,127,156)(17,120,50)(18,51,117)(19,118,52)(20,49,119)(21,124,163)(22,164,121)(23,122,161)(24,162,123)(25,115,188)(26,185,116)(27,113,186)(28,187,114)(29,174,96)(30,93,175)(31,176,94)(32,95,173)(33,73,106)(34,107,74)(35,75,108)(36,105,76)(37,148,131)(38,132,145)(39,146,129)(40,130,147)(41,152,135)(42,136,149)(43,150,133)(44,134,151)(45,55,144)(46,141,56)(47,53,142)(48,143,54)(57,90,184)(58,181,91)(59,92,182)(60,183,89)(61,201,137)(62,138,202)(63,203,139)(64,140,204)(65,189,86)(66,87,190)(67,191,88)(68,85,192)(69,172,197)(70,198,169)(71,170,199)(72,200,171)(81,214,208)(82,205,215)(83,216,206)(84,207,213)(157,179,196)(158,193,180)(159,177,194)(160,195,178), (1,49,201)(2,202,50)(3,51,203)(4,204,52)(5,72,155)(6,156,69)(7,70,153)(8,154,71)(9,93,130)(10,131,94)(11,95,132)(12,129,96)(13,97,169)(14,170,98)(15,99,171)(16,172,100)(17,109,62)(18,63,110)(19,111,64)(20,61,112)(21,28,87)(22,88,25)(23,26,85)(24,86,27)(29,102,39)(30,40,103)(31,104,37)(32,38,101)(33,133,178)(34,179,134)(35,135,180)(36,177,136)(41,158,75)(42,76,159)(43,160,73)(44,74,157)(45,90,213)(46,214,91)(47,92,215)(48,216,89)(53,182,82)(54,83,183)(55,184,84)(56,81,181)(57,207,144)(58,141,208)(59,205,142)(60,143,206)(65,113,162)(66,163,114)(67,115,164)(68,161,116)(77,125,199)(78,200,126)(79,127,197)(80,198,128)(105,194,149)(106,150,195)(107,196,151)(108,152,193)(117,139,209)(118,210,140)(119,137,211)(120,212,138)(121,191,188)(122,185,192)(123,189,186)(124,187,190)(145,167,173)(146,174,168)(147,165,175)(148,176,166), (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)(193,194,195,196)(197,198,199,200)(201,202,203,204)(205,206,207,208)(209,210,211,212)(213,214,215,216), (1,9)(2,12)(3,11)(4,10)(5,84)(6,83)(7,82)(8,81)(13,142)(14,141)(15,144)(16,143)(17,146)(18,145)(19,148)(20,147)(21,150)(22,149)(23,152)(24,151)(25,105)(26,108)(27,107)(28,106)(29,138)(30,137)(31,140)(32,139)(33,114)(34,113)(35,116)(36,115)(37,118)(38,117)(39,120)(40,119)(41,122)(42,121)(43,124)(44,123)(45,126)(46,125)(47,128)(48,127)(49,130)(50,129)(51,132)(52,131)(53,153)(54,156)(55,155)(56,154)(57,171)(58,170)(59,169)(60,172)(61,175)(62,174)(63,173)(64,176)(65,179)(66,178)(67,177)(68,180)(69,183)(70,182)(71,181)(72,184)(73,187)(74,186)(75,185)(76,188)(77,214)(78,213)(79,216)(80,215)(85,193)(86,196)(87,195)(88,194)(89,197)(90,200)(91,199)(92,198)(93,201)(94,204)(95,203)(96,202)(97,205)(98,208)(99,207)(100,206)(101,209)(102,212)(103,211)(104,210)(109,168)(110,167)(111,166)(112,165)(133,163)(134,162)(135,161)(136,164)(157,189)(158,192)(159,191)(160,190)>;
G:=Group( (1,10)(2,11)(3,12)(4,9)(5,83)(6,84)(7,81)(8,82)(13,141)(14,142)(15,143)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,106)(26,107)(27,108)(28,105)(29,139)(30,140)(31,137)(32,138)(33,115)(34,116)(35,113)(36,114)(37,119)(38,120)(39,117)(40,118)(41,123)(42,124)(43,121)(44,122)(45,127)(46,128)(47,125)(48,126)(49,131)(50,132)(51,129)(52,130)(53,154)(54,155)(55,156)(56,153)(57,172)(58,169)(59,170)(60,171)(61,176)(62,173)(63,174)(64,175)(65,180)(66,177)(67,178)(68,179)(69,184)(70,181)(71,182)(72,183)(73,188)(74,185)(75,186)(76,187)(77,215)(78,216)(79,213)(80,214)(85,196)(86,193)(87,194)(88,195)(89,200)(90,197)(91,198)(92,199)(93,204)(94,201)(95,202)(96,203)(97,208)(98,205)(99,206)(100,207)(101,212)(102,209)(103,210)(104,211)(109,167)(110,168)(111,165)(112,166)(133,164)(134,161)(135,162)(136,163)(157,192)(158,189)(159,190)(160,191), (1,162,182)(2,183,163)(3,164,184)(4,181,161)(5,177,132)(6,129,178)(7,179,130)(8,131,180)(9,70,134)(10,135,71)(11,72,136)(12,133,69)(13,74,30)(14,31,75)(15,76,32)(16,29,73)(17,216,87)(18,88,213)(19,214,85)(20,86,215)(21,109,89)(22,90,110)(23,111,91)(24,92,112)(25,45,63)(26,64,46)(27,47,61)(28,62,48)(33,156,96)(34,93,153)(35,154,94)(36,95,155)(37,158,98)(38,99,159)(39,160,100)(40,97,157)(41,170,104)(42,101,171)(43,172,102)(44,103,169)(49,65,82)(50,83,66)(51,67,84)(52,81,68)(53,201,113)(54,114,202)(55,203,115)(56,116,204)(57,209,121)(58,122,210)(59,211,123)(60,124,212)(77,148,193)(78,194,145)(79,146,195)(80,196,147)(105,173,126)(106,127,174)(107,175,128)(108,125,176)(117,191,207)(118,208,192)(119,189,205)(120,206,190)(137,186,142)(138,143,187)(139,188,144)(140,141,185)(149,167,200)(150,197,168)(151,165,198)(152,199,166), (1,211,112)(2,109,212)(3,209,110)(4,111,210)(5,78,99)(6,100,79)(7,80,97)(8,98,77)(9,165,103)(10,104,166)(11,167,101)(12,102,168)(13,153,128)(14,125,154)(15,155,126)(16,127,156)(17,120,50)(18,51,117)(19,118,52)(20,49,119)(21,124,163)(22,164,121)(23,122,161)(24,162,123)(25,115,188)(26,185,116)(27,113,186)(28,187,114)(29,174,96)(30,93,175)(31,176,94)(32,95,173)(33,73,106)(34,107,74)(35,75,108)(36,105,76)(37,148,131)(38,132,145)(39,146,129)(40,130,147)(41,152,135)(42,136,149)(43,150,133)(44,134,151)(45,55,144)(46,141,56)(47,53,142)(48,143,54)(57,90,184)(58,181,91)(59,92,182)(60,183,89)(61,201,137)(62,138,202)(63,203,139)(64,140,204)(65,189,86)(66,87,190)(67,191,88)(68,85,192)(69,172,197)(70,198,169)(71,170,199)(72,200,171)(81,214,208)(82,205,215)(83,216,206)(84,207,213)(157,179,196)(158,193,180)(159,177,194)(160,195,178), (1,49,201)(2,202,50)(3,51,203)(4,204,52)(5,72,155)(6,156,69)(7,70,153)(8,154,71)(9,93,130)(10,131,94)(11,95,132)(12,129,96)(13,97,169)(14,170,98)(15,99,171)(16,172,100)(17,109,62)(18,63,110)(19,111,64)(20,61,112)(21,28,87)(22,88,25)(23,26,85)(24,86,27)(29,102,39)(30,40,103)(31,104,37)(32,38,101)(33,133,178)(34,179,134)(35,135,180)(36,177,136)(41,158,75)(42,76,159)(43,160,73)(44,74,157)(45,90,213)(46,214,91)(47,92,215)(48,216,89)(53,182,82)(54,83,183)(55,184,84)(56,81,181)(57,207,144)(58,141,208)(59,205,142)(60,143,206)(65,113,162)(66,163,114)(67,115,164)(68,161,116)(77,125,199)(78,200,126)(79,127,197)(80,198,128)(105,194,149)(106,150,195)(107,196,151)(108,152,193)(117,139,209)(118,210,140)(119,137,211)(120,212,138)(121,191,188)(122,185,192)(123,189,186)(124,187,190)(145,167,173)(146,174,168)(147,165,175)(148,176,166), (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)(193,194,195,196)(197,198,199,200)(201,202,203,204)(205,206,207,208)(209,210,211,212)(213,214,215,216), (1,9)(2,12)(3,11)(4,10)(5,84)(6,83)(7,82)(8,81)(13,142)(14,141)(15,144)(16,143)(17,146)(18,145)(19,148)(20,147)(21,150)(22,149)(23,152)(24,151)(25,105)(26,108)(27,107)(28,106)(29,138)(30,137)(31,140)(32,139)(33,114)(34,113)(35,116)(36,115)(37,118)(38,117)(39,120)(40,119)(41,122)(42,121)(43,124)(44,123)(45,126)(46,125)(47,128)(48,127)(49,130)(50,129)(51,132)(52,131)(53,153)(54,156)(55,155)(56,154)(57,171)(58,170)(59,169)(60,172)(61,175)(62,174)(63,173)(64,176)(65,179)(66,178)(67,177)(68,180)(69,183)(70,182)(71,181)(72,184)(73,187)(74,186)(75,185)(76,188)(77,214)(78,213)(79,216)(80,215)(85,193)(86,196)(87,195)(88,194)(89,197)(90,200)(91,199)(92,198)(93,201)(94,204)(95,203)(96,202)(97,205)(98,208)(99,207)(100,206)(101,209)(102,212)(103,211)(104,210)(109,168)(110,167)(111,166)(112,165)(133,163)(134,162)(135,161)(136,164)(157,189)(158,192)(159,191)(160,190) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,9),(5,83),(6,84),(7,81),(8,82),(13,141),(14,142),(15,143),(16,144),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,152),(25,106),(26,107),(27,108),(28,105),(29,139),(30,140),(31,137),(32,138),(33,115),(34,116),(35,113),(36,114),(37,119),(38,120),(39,117),(40,118),(41,123),(42,124),(43,121),(44,122),(45,127),(46,128),(47,125),(48,126),(49,131),(50,132),(51,129),(52,130),(53,154),(54,155),(55,156),(56,153),(57,172),(58,169),(59,170),(60,171),(61,176),(62,173),(63,174),(64,175),(65,180),(66,177),(67,178),(68,179),(69,184),(70,181),(71,182),(72,183),(73,188),(74,185),(75,186),(76,187),(77,215),(78,216),(79,213),(80,214),(85,196),(86,193),(87,194),(88,195),(89,200),(90,197),(91,198),(92,199),(93,204),(94,201),(95,202),(96,203),(97,208),(98,205),(99,206),(100,207),(101,212),(102,209),(103,210),(104,211),(109,167),(110,168),(111,165),(112,166),(133,164),(134,161),(135,162),(136,163),(157,192),(158,189),(159,190),(160,191)], [(1,162,182),(2,183,163),(3,164,184),(4,181,161),(5,177,132),(6,129,178),(7,179,130),(8,131,180),(9,70,134),(10,135,71),(11,72,136),(12,133,69),(13,74,30),(14,31,75),(15,76,32),(16,29,73),(17,216,87),(18,88,213),(19,214,85),(20,86,215),(21,109,89),(22,90,110),(23,111,91),(24,92,112),(25,45,63),(26,64,46),(27,47,61),(28,62,48),(33,156,96),(34,93,153),(35,154,94),(36,95,155),(37,158,98),(38,99,159),(39,160,100),(40,97,157),(41,170,104),(42,101,171),(43,172,102),(44,103,169),(49,65,82),(50,83,66),(51,67,84),(52,81,68),(53,201,113),(54,114,202),(55,203,115),(56,116,204),(57,209,121),(58,122,210),(59,211,123),(60,124,212),(77,148,193),(78,194,145),(79,146,195),(80,196,147),(105,173,126),(106,127,174),(107,175,128),(108,125,176),(117,191,207),(118,208,192),(119,189,205),(120,206,190),(137,186,142),(138,143,187),(139,188,144),(140,141,185),(149,167,200),(150,197,168),(151,165,198),(152,199,166)], [(1,211,112),(2,109,212),(3,209,110),(4,111,210),(5,78,99),(6,100,79),(7,80,97),(8,98,77),(9,165,103),(10,104,166),(11,167,101),(12,102,168),(13,153,128),(14,125,154),(15,155,126),(16,127,156),(17,120,50),(18,51,117),(19,118,52),(20,49,119),(21,124,163),(22,164,121),(23,122,161),(24,162,123),(25,115,188),(26,185,116),(27,113,186),(28,187,114),(29,174,96),(30,93,175),(31,176,94),(32,95,173),(33,73,106),(34,107,74),(35,75,108),(36,105,76),(37,148,131),(38,132,145),(39,146,129),(40,130,147),(41,152,135),(42,136,149),(43,150,133),(44,134,151),(45,55,144),(46,141,56),(47,53,142),(48,143,54),(57,90,184),(58,181,91),(59,92,182),(60,183,89),(61,201,137),(62,138,202),(63,203,139),(64,140,204),(65,189,86),(66,87,190),(67,191,88),(68,85,192),(69,172,197),(70,198,169),(71,170,199),(72,200,171),(81,214,208),(82,205,215),(83,216,206),(84,207,213),(157,179,196),(158,193,180),(159,177,194),(160,195,178)], [(1,49,201),(2,202,50),(3,51,203),(4,204,52),(5,72,155),(6,156,69),(7,70,153),(8,154,71),(9,93,130),(10,131,94),(11,95,132),(12,129,96),(13,97,169),(14,170,98),(15,99,171),(16,172,100),(17,109,62),(18,63,110),(19,111,64),(20,61,112),(21,28,87),(22,88,25),(23,26,85),(24,86,27),(29,102,39),(30,40,103),(31,104,37),(32,38,101),(33,133,178),(34,179,134),(35,135,180),(36,177,136),(41,158,75),(42,76,159),(43,160,73),(44,74,157),(45,90,213),(46,214,91),(47,92,215),(48,216,89),(53,182,82),(54,83,183),(55,184,84),(56,81,181),(57,207,144),(58,141,208),(59,205,142),(60,143,206),(65,113,162),(66,163,114),(67,115,164),(68,161,116),(77,125,199),(78,200,126),(79,127,197),(80,198,128),(105,194,149),(106,150,195),(107,196,151),(108,152,193),(117,139,209),(118,210,140),(119,137,211),(120,212,138),(121,191,188),(122,185,192),(123,189,186),(124,187,190),(145,167,173),(146,174,168),(147,165,175),(148,176,166)], [(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),(193,194,195,196),(197,198,199,200),(201,202,203,204),(205,206,207,208),(209,210,211,212),(213,214,215,216)], [(1,9),(2,12),(3,11),(4,10),(5,84),(6,83),(7,82),(8,81),(13,142),(14,141),(15,144),(16,143),(17,146),(18,145),(19,148),(20,147),(21,150),(22,149),(23,152),(24,151),(25,105),(26,108),(27,107),(28,106),(29,138),(30,137),(31,140),(32,139),(33,114),(34,113),(35,116),(36,115),(37,118),(38,117),(39,120),(40,119),(41,122),(42,121),(43,124),(44,123),(45,126),(46,125),(47,128),(48,127),(49,130),(50,129),(51,132),(52,131),(53,153),(54,156),(55,155),(56,154),(57,171),(58,170),(59,169),(60,172),(61,175),(62,174),(63,173),(64,176),(65,179),(66,178),(67,177),(68,180),(69,183),(70,182),(71,181),(72,184),(73,187),(74,186),(75,185),(76,188),(77,214),(78,213),(79,216),(80,215),(85,193),(86,196),(87,195),(88,194),(89,197),(90,200),(91,199),(92,198),(93,201),(94,204),(95,203),(96,202),(97,205),(98,208),(99,207),(100,206),(101,209),(102,212),(103,211),(104,210),(109,168),(110,167),(111,166),(112,165),(133,163),(134,162),(135,161),(136,164),(157,189),(158,192),(159,191),(160,190)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3M | 4A | 4B | 6A | ··· | 6CM |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 54 | 54 | 2 | ··· | 2 | 54 | 54 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 |
| kernel | C2×C33⋊15D4 | C2×C33⋊5C4 | C33⋊15D4 | C22×C33⋊C2 | C63 | C2×C62 | C32×C6 | C62 | C3×C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 13 | 2 | 39 | 52 |
Matrix representation of C2×C33⋊15D4 ►in GL6(𝔽13)
| 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 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 1 | 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 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 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 | 1 | 0 |
G:=sub<GL(6,GF(13))| [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],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2×C33⋊15D4 in GAP, Magma, Sage, TeX
C_2\times C_3^3\rtimes_{15}D_4 % in TeX
G:=Group("C2xC3^3:15D4"); // GroupNames label
G:=SmallGroup(432,729);
// by ID
G=gap.SmallGroup(432,729);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,1124,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=f*b*f=b^-1,c*d=d*c,e*c*e^-1=f*c*f=c^-1,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations