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## G = C2×C4×C33⋊C2order 432 = 24·33

### Direct product of C2×C4 and C33⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C4×C33⋊C2
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C2×C33⋊C2 — C22×C33⋊C2 — C2×C4×C33⋊C2
 Lower central C33 — C2×C4×C33⋊C2
 Upper central C1 — C2×C4

Generators and relations for C2×C4×C33⋊C2
G = < a,b,c,d,e,f | a2=b4=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=c-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 3720 in 756 conjugacy classes, 243 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C22×C4, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C33, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C2×C4, C33⋊C2, C32×C6, C32×C6, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C335C4, C32×C12, C2×C33⋊C2, C3×C62, C2×C4×C3⋊S3, C4×C33⋊C2, C2×C335C4, C3×C6×C12, C22×C33⋊C2, C2×C4×C33⋊C2
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C3⋊S3, C4×S3, C22×S3, C2×C3⋊S3, S3×C2×C4, C33⋊C2, C4×C3⋊S3, C22×C3⋊S3, C2×C33⋊C2, C2×C4×C3⋊S3, C4×C33⋊C2, C22×C33⋊C2, C2×C4×C33⋊C2

Smallest permutation representation of C2×C4×C33⋊C2
On 216 points
Generators in S216
(1 148)(2 145)(3 146)(4 147)(5 143)(6 144)(7 141)(8 142)(9 198)(10 199)(11 200)(12 197)(13 202)(14 203)(15 204)(16 201)(17 206)(18 207)(19 208)(20 205)(21 210)(22 211)(23 212)(24 209)(25 166)(26 167)(27 168)(28 165)(29 115)(30 116)(31 113)(32 114)(33 176)(34 173)(35 174)(36 175)(37 180)(38 177)(39 178)(40 179)(41 184)(42 181)(43 182)(44 183)(45 188)(46 185)(47 186)(48 187)(49 192)(50 189)(51 190)(52 191)(53 109)(54 110)(55 111)(56 112)(57 169)(58 170)(59 171)(60 172)(61 193)(62 194)(63 195)(64 196)(65 123)(66 124)(67 121)(68 122)(69 127)(70 128)(71 125)(72 126)(73 131)(74 132)(75 129)(76 130)(77 135)(78 136)(79 133)(80 134)(81 137)(82 138)(83 139)(84 140)(85 119)(86 120)(87 117)(88 118)(89 151)(90 152)(91 149)(92 150)(93 155)(94 156)(95 153)(96 154)(97 159)(98 160)(99 157)(100 158)(101 163)(102 164)(103 161)(104 162)(105 213)(106 214)(107 215)(108 216)
(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)
(1 110 20)(2 111 17)(3 112 18)(4 109 19)(5 77 14)(6 78 15)(7 79 16)(8 80 13)(9 168 74)(10 165 75)(11 166 76)(12 167 73)(21 91 58)(22 92 59)(23 89 60)(24 90 57)(25 130 200)(26 131 197)(27 132 198)(28 129 199)(29 97 40)(30 98 37)(31 99 38)(32 100 39)(33 106 96)(34 107 93)(35 108 94)(36 105 95)(41 87 101)(42 88 102)(43 85 103)(44 86 104)(45 65 139)(46 66 140)(47 67 137)(48 68 138)(49 71 64)(50 72 61)(51 69 62)(52 70 63)(53 208 147)(54 205 148)(55 206 145)(56 207 146)(81 186 121)(82 187 122)(83 188 123)(84 185 124)(113 157 177)(114 158 178)(115 159 179)(116 160 180)(117 163 184)(118 164 181)(119 161 182)(120 162 183)(125 196 192)(126 193 189)(127 194 190)(128 195 191)(133 201 141)(134 202 142)(135 203 143)(136 204 144)(149 170 210)(150 171 211)(151 172 212)(152 169 209)(153 175 213)(154 176 214)(155 173 215)(156 174 216)
(1 49 29)(2 50 30)(3 51 31)(4 52 32)(5 48 108)(6 45 105)(7 46 106)(8 47 107)(9 43 57)(10 44 58)(11 41 59)(12 42 60)(13 137 34)(14 138 35)(15 139 36)(16 140 33)(17 61 37)(18 62 38)(19 63 39)(20 64 40)(21 165 86)(22 166 87)(23 167 88)(24 168 85)(25 117 211)(26 118 212)(27 119 209)(28 120 210)(53 128 158)(54 125 159)(55 126 160)(56 127 157)(65 95 78)(66 96 79)(67 93 80)(68 94 77)(69 99 112)(70 100 109)(71 97 110)(72 98 111)(73 102 89)(74 103 90)(75 104 91)(76 101 92)(81 173 202)(82 174 203)(83 175 204)(84 176 201)(113 146 190)(114 147 191)(115 148 192)(116 145 189)(121 155 134)(122 156 135)(123 153 136)(124 154 133)(129 162 149)(130 163 150)(131 164 151)(132 161 152)(141 185 214)(142 186 215)(143 187 216)(144 188 213)(169 198 182)(170 199 183)(171 200 184)(172 197 181)(177 207 194)(178 208 195)(179 205 196)(180 206 193)
(1 80 43)(2 77 44)(3 78 41)(4 79 42)(5 104 17)(6 101 18)(7 102 19)(8 103 20)(9 29 93)(10 30 94)(11 31 95)(12 32 96)(13 85 110)(14 86 111)(15 87 112)(16 88 109)(21 72 138)(22 69 139)(23 70 140)(24 71 137)(25 157 175)(26 158 176)(27 159 173)(28 160 174)(33 167 100)(34 168 97)(35 165 98)(36 166 99)(37 108 75)(38 105 76)(39 106 73)(40 107 74)(45 92 62)(46 89 63)(47 90 64)(48 91 61)(49 67 57)(50 68 58)(51 65 59)(52 66 60)(53 201 118)(54 202 119)(55 203 120)(56 204 117)(81 209 125)(82 210 126)(83 211 127)(84 212 128)(113 153 200)(114 154 197)(115 155 198)(116 156 199)(121 169 192)(122 170 189)(123 171 190)(124 172 191)(129 180 216)(130 177 213)(131 178 214)(132 179 215)(133 181 147)(134 182 148)(135 183 145)(136 184 146)(141 164 208)(142 161 205)(143 162 206)(144 163 207)(149 193 187)(150 194 188)(151 195 185)(152 196 186)
(1 148)(2 145)(3 146)(4 147)(5 120)(6 117)(7 118)(8 119)(9 121)(10 122)(11 123)(12 124)(13 161)(14 162)(15 163)(16 164)(17 55)(18 56)(19 53)(20 54)(21 216)(22 213)(23 214)(24 215)(25 45)(26 46)(27 47)(28 48)(29 192)(30 189)(31 190)(32 191)(33 151)(34 152)(35 149)(36 150)(37 126)(38 127)(39 128)(40 125)(41 136)(42 133)(43 134)(44 135)(49 115)(50 116)(51 113)(52 114)(57 155)(58 156)(59 153)(60 154)(61 160)(62 157)(63 158)(64 159)(65 200)(66 197)(67 198)(68 199)(69 177)(70 178)(71 179)(72 180)(73 84)(74 81)(75 82)(76 83)(77 183)(78 184)(79 181)(80 182)(85 142)(86 143)(87 144)(88 141)(89 176)(90 173)(91 174)(92 175)(93 169)(94 170)(95 171)(96 172)(97 196)(98 193)(99 194)(100 195)(101 204)(102 201)(103 202)(104 203)(105 211)(106 212)(107 209)(108 210)(109 208)(110 205)(111 206)(112 207)(129 138)(130 139)(131 140)(132 137)(165 187)(166 188)(167 185)(168 186)

G:=sub<Sym(216)| (1,148)(2,145)(3,146)(4,147)(5,143)(6,144)(7,141)(8,142)(9,198)(10,199)(11,200)(12,197)(13,202)(14,203)(15,204)(16,201)(17,206)(18,207)(19,208)(20,205)(21,210)(22,211)(23,212)(24,209)(25,166)(26,167)(27,168)(28,165)(29,115)(30,116)(31,113)(32,114)(33,176)(34,173)(35,174)(36,175)(37,180)(38,177)(39,178)(40,179)(41,184)(42,181)(43,182)(44,183)(45,188)(46,185)(47,186)(48,187)(49,192)(50,189)(51,190)(52,191)(53,109)(54,110)(55,111)(56,112)(57,169)(58,170)(59,171)(60,172)(61,193)(62,194)(63,195)(64,196)(65,123)(66,124)(67,121)(68,122)(69,127)(70,128)(71,125)(72,126)(73,131)(74,132)(75,129)(76,130)(77,135)(78,136)(79,133)(80,134)(81,137)(82,138)(83,139)(84,140)(85,119)(86,120)(87,117)(88,118)(89,151)(90,152)(91,149)(92,150)(93,155)(94,156)(95,153)(96,154)(97,159)(98,160)(99,157)(100,158)(101,163)(102,164)(103,161)(104,162)(105,213)(106,214)(107,215)(108,216), (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), (1,110,20)(2,111,17)(3,112,18)(4,109,19)(5,77,14)(6,78,15)(7,79,16)(8,80,13)(9,168,74)(10,165,75)(11,166,76)(12,167,73)(21,91,58)(22,92,59)(23,89,60)(24,90,57)(25,130,200)(26,131,197)(27,132,198)(28,129,199)(29,97,40)(30,98,37)(31,99,38)(32,100,39)(33,106,96)(34,107,93)(35,108,94)(36,105,95)(41,87,101)(42,88,102)(43,85,103)(44,86,104)(45,65,139)(46,66,140)(47,67,137)(48,68,138)(49,71,64)(50,72,61)(51,69,62)(52,70,63)(53,208,147)(54,205,148)(55,206,145)(56,207,146)(81,186,121)(82,187,122)(83,188,123)(84,185,124)(113,157,177)(114,158,178)(115,159,179)(116,160,180)(117,163,184)(118,164,181)(119,161,182)(120,162,183)(125,196,192)(126,193,189)(127,194,190)(128,195,191)(133,201,141)(134,202,142)(135,203,143)(136,204,144)(149,170,210)(150,171,211)(151,172,212)(152,169,209)(153,175,213)(154,176,214)(155,173,215)(156,174,216), (1,49,29)(2,50,30)(3,51,31)(4,52,32)(5,48,108)(6,45,105)(7,46,106)(8,47,107)(9,43,57)(10,44,58)(11,41,59)(12,42,60)(13,137,34)(14,138,35)(15,139,36)(16,140,33)(17,61,37)(18,62,38)(19,63,39)(20,64,40)(21,165,86)(22,166,87)(23,167,88)(24,168,85)(25,117,211)(26,118,212)(27,119,209)(28,120,210)(53,128,158)(54,125,159)(55,126,160)(56,127,157)(65,95,78)(66,96,79)(67,93,80)(68,94,77)(69,99,112)(70,100,109)(71,97,110)(72,98,111)(73,102,89)(74,103,90)(75,104,91)(76,101,92)(81,173,202)(82,174,203)(83,175,204)(84,176,201)(113,146,190)(114,147,191)(115,148,192)(116,145,189)(121,155,134)(122,156,135)(123,153,136)(124,154,133)(129,162,149)(130,163,150)(131,164,151)(132,161,152)(141,185,214)(142,186,215)(143,187,216)(144,188,213)(169,198,182)(170,199,183)(171,200,184)(172,197,181)(177,207,194)(178,208,195)(179,205,196)(180,206,193), (1,80,43)(2,77,44)(3,78,41)(4,79,42)(5,104,17)(6,101,18)(7,102,19)(8,103,20)(9,29,93)(10,30,94)(11,31,95)(12,32,96)(13,85,110)(14,86,111)(15,87,112)(16,88,109)(21,72,138)(22,69,139)(23,70,140)(24,71,137)(25,157,175)(26,158,176)(27,159,173)(28,160,174)(33,167,100)(34,168,97)(35,165,98)(36,166,99)(37,108,75)(38,105,76)(39,106,73)(40,107,74)(45,92,62)(46,89,63)(47,90,64)(48,91,61)(49,67,57)(50,68,58)(51,65,59)(52,66,60)(53,201,118)(54,202,119)(55,203,120)(56,204,117)(81,209,125)(82,210,126)(83,211,127)(84,212,128)(113,153,200)(114,154,197)(115,155,198)(116,156,199)(121,169,192)(122,170,189)(123,171,190)(124,172,191)(129,180,216)(130,177,213)(131,178,214)(132,179,215)(133,181,147)(134,182,148)(135,183,145)(136,184,146)(141,164,208)(142,161,205)(143,162,206)(144,163,207)(149,193,187)(150,194,188)(151,195,185)(152,196,186), (1,148)(2,145)(3,146)(4,147)(5,120)(6,117)(7,118)(8,119)(9,121)(10,122)(11,123)(12,124)(13,161)(14,162)(15,163)(16,164)(17,55)(18,56)(19,53)(20,54)(21,216)(22,213)(23,214)(24,215)(25,45)(26,46)(27,47)(28,48)(29,192)(30,189)(31,190)(32,191)(33,151)(34,152)(35,149)(36,150)(37,126)(38,127)(39,128)(40,125)(41,136)(42,133)(43,134)(44,135)(49,115)(50,116)(51,113)(52,114)(57,155)(58,156)(59,153)(60,154)(61,160)(62,157)(63,158)(64,159)(65,200)(66,197)(67,198)(68,199)(69,177)(70,178)(71,179)(72,180)(73,84)(74,81)(75,82)(76,83)(77,183)(78,184)(79,181)(80,182)(85,142)(86,143)(87,144)(88,141)(89,176)(90,173)(91,174)(92,175)(93,169)(94,170)(95,171)(96,172)(97,196)(98,193)(99,194)(100,195)(101,204)(102,201)(103,202)(104,203)(105,211)(106,212)(107,209)(108,210)(109,208)(110,205)(111,206)(112,207)(129,138)(130,139)(131,140)(132,137)(165,187)(166,188)(167,185)(168,186)>;

G:=Group( (1,148)(2,145)(3,146)(4,147)(5,143)(6,144)(7,141)(8,142)(9,198)(10,199)(11,200)(12,197)(13,202)(14,203)(15,204)(16,201)(17,206)(18,207)(19,208)(20,205)(21,210)(22,211)(23,212)(24,209)(25,166)(26,167)(27,168)(28,165)(29,115)(30,116)(31,113)(32,114)(33,176)(34,173)(35,174)(36,175)(37,180)(38,177)(39,178)(40,179)(41,184)(42,181)(43,182)(44,183)(45,188)(46,185)(47,186)(48,187)(49,192)(50,189)(51,190)(52,191)(53,109)(54,110)(55,111)(56,112)(57,169)(58,170)(59,171)(60,172)(61,193)(62,194)(63,195)(64,196)(65,123)(66,124)(67,121)(68,122)(69,127)(70,128)(71,125)(72,126)(73,131)(74,132)(75,129)(76,130)(77,135)(78,136)(79,133)(80,134)(81,137)(82,138)(83,139)(84,140)(85,119)(86,120)(87,117)(88,118)(89,151)(90,152)(91,149)(92,150)(93,155)(94,156)(95,153)(96,154)(97,159)(98,160)(99,157)(100,158)(101,163)(102,164)(103,161)(104,162)(105,213)(106,214)(107,215)(108,216), (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), (1,110,20)(2,111,17)(3,112,18)(4,109,19)(5,77,14)(6,78,15)(7,79,16)(8,80,13)(9,168,74)(10,165,75)(11,166,76)(12,167,73)(21,91,58)(22,92,59)(23,89,60)(24,90,57)(25,130,200)(26,131,197)(27,132,198)(28,129,199)(29,97,40)(30,98,37)(31,99,38)(32,100,39)(33,106,96)(34,107,93)(35,108,94)(36,105,95)(41,87,101)(42,88,102)(43,85,103)(44,86,104)(45,65,139)(46,66,140)(47,67,137)(48,68,138)(49,71,64)(50,72,61)(51,69,62)(52,70,63)(53,208,147)(54,205,148)(55,206,145)(56,207,146)(81,186,121)(82,187,122)(83,188,123)(84,185,124)(113,157,177)(114,158,178)(115,159,179)(116,160,180)(117,163,184)(118,164,181)(119,161,182)(120,162,183)(125,196,192)(126,193,189)(127,194,190)(128,195,191)(133,201,141)(134,202,142)(135,203,143)(136,204,144)(149,170,210)(150,171,211)(151,172,212)(152,169,209)(153,175,213)(154,176,214)(155,173,215)(156,174,216), (1,49,29)(2,50,30)(3,51,31)(4,52,32)(5,48,108)(6,45,105)(7,46,106)(8,47,107)(9,43,57)(10,44,58)(11,41,59)(12,42,60)(13,137,34)(14,138,35)(15,139,36)(16,140,33)(17,61,37)(18,62,38)(19,63,39)(20,64,40)(21,165,86)(22,166,87)(23,167,88)(24,168,85)(25,117,211)(26,118,212)(27,119,209)(28,120,210)(53,128,158)(54,125,159)(55,126,160)(56,127,157)(65,95,78)(66,96,79)(67,93,80)(68,94,77)(69,99,112)(70,100,109)(71,97,110)(72,98,111)(73,102,89)(74,103,90)(75,104,91)(76,101,92)(81,173,202)(82,174,203)(83,175,204)(84,176,201)(113,146,190)(114,147,191)(115,148,192)(116,145,189)(121,155,134)(122,156,135)(123,153,136)(124,154,133)(129,162,149)(130,163,150)(131,164,151)(132,161,152)(141,185,214)(142,186,215)(143,187,216)(144,188,213)(169,198,182)(170,199,183)(171,200,184)(172,197,181)(177,207,194)(178,208,195)(179,205,196)(180,206,193), (1,80,43)(2,77,44)(3,78,41)(4,79,42)(5,104,17)(6,101,18)(7,102,19)(8,103,20)(9,29,93)(10,30,94)(11,31,95)(12,32,96)(13,85,110)(14,86,111)(15,87,112)(16,88,109)(21,72,138)(22,69,139)(23,70,140)(24,71,137)(25,157,175)(26,158,176)(27,159,173)(28,160,174)(33,167,100)(34,168,97)(35,165,98)(36,166,99)(37,108,75)(38,105,76)(39,106,73)(40,107,74)(45,92,62)(46,89,63)(47,90,64)(48,91,61)(49,67,57)(50,68,58)(51,65,59)(52,66,60)(53,201,118)(54,202,119)(55,203,120)(56,204,117)(81,209,125)(82,210,126)(83,211,127)(84,212,128)(113,153,200)(114,154,197)(115,155,198)(116,156,199)(121,169,192)(122,170,189)(123,171,190)(124,172,191)(129,180,216)(130,177,213)(131,178,214)(132,179,215)(133,181,147)(134,182,148)(135,183,145)(136,184,146)(141,164,208)(142,161,205)(143,162,206)(144,163,207)(149,193,187)(150,194,188)(151,195,185)(152,196,186), (1,148)(2,145)(3,146)(4,147)(5,120)(6,117)(7,118)(8,119)(9,121)(10,122)(11,123)(12,124)(13,161)(14,162)(15,163)(16,164)(17,55)(18,56)(19,53)(20,54)(21,216)(22,213)(23,214)(24,215)(25,45)(26,46)(27,47)(28,48)(29,192)(30,189)(31,190)(32,191)(33,151)(34,152)(35,149)(36,150)(37,126)(38,127)(39,128)(40,125)(41,136)(42,133)(43,134)(44,135)(49,115)(50,116)(51,113)(52,114)(57,155)(58,156)(59,153)(60,154)(61,160)(62,157)(63,158)(64,159)(65,200)(66,197)(67,198)(68,199)(69,177)(70,178)(71,179)(72,180)(73,84)(74,81)(75,82)(76,83)(77,183)(78,184)(79,181)(80,182)(85,142)(86,143)(87,144)(88,141)(89,176)(90,173)(91,174)(92,175)(93,169)(94,170)(95,171)(96,172)(97,196)(98,193)(99,194)(100,195)(101,204)(102,201)(103,202)(104,203)(105,211)(106,212)(107,209)(108,210)(109,208)(110,205)(111,206)(112,207)(129,138)(130,139)(131,140)(132,137)(165,187)(166,188)(167,185)(168,186) );

G=PermutationGroup([[(1,148),(2,145),(3,146),(4,147),(5,143),(6,144),(7,141),(8,142),(9,198),(10,199),(11,200),(12,197),(13,202),(14,203),(15,204),(16,201),(17,206),(18,207),(19,208),(20,205),(21,210),(22,211),(23,212),(24,209),(25,166),(26,167),(27,168),(28,165),(29,115),(30,116),(31,113),(32,114),(33,176),(34,173),(35,174),(36,175),(37,180),(38,177),(39,178),(40,179),(41,184),(42,181),(43,182),(44,183),(45,188),(46,185),(47,186),(48,187),(49,192),(50,189),(51,190),(52,191),(53,109),(54,110),(55,111),(56,112),(57,169),(58,170),(59,171),(60,172),(61,193),(62,194),(63,195),(64,196),(65,123),(66,124),(67,121),(68,122),(69,127),(70,128),(71,125),(72,126),(73,131),(74,132),(75,129),(76,130),(77,135),(78,136),(79,133),(80,134),(81,137),(82,138),(83,139),(84,140),(85,119),(86,120),(87,117),(88,118),(89,151),(90,152),(91,149),(92,150),(93,155),(94,156),(95,153),(96,154),(97,159),(98,160),(99,157),(100,158),(101,163),(102,164),(103,161),(104,162),(105,213),(106,214),(107,215),(108,216)], [(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)], [(1,110,20),(2,111,17),(3,112,18),(4,109,19),(5,77,14),(6,78,15),(7,79,16),(8,80,13),(9,168,74),(10,165,75),(11,166,76),(12,167,73),(21,91,58),(22,92,59),(23,89,60),(24,90,57),(25,130,200),(26,131,197),(27,132,198),(28,129,199),(29,97,40),(30,98,37),(31,99,38),(32,100,39),(33,106,96),(34,107,93),(35,108,94),(36,105,95),(41,87,101),(42,88,102),(43,85,103),(44,86,104),(45,65,139),(46,66,140),(47,67,137),(48,68,138),(49,71,64),(50,72,61),(51,69,62),(52,70,63),(53,208,147),(54,205,148),(55,206,145),(56,207,146),(81,186,121),(82,187,122),(83,188,123),(84,185,124),(113,157,177),(114,158,178),(115,159,179),(116,160,180),(117,163,184),(118,164,181),(119,161,182),(120,162,183),(125,196,192),(126,193,189),(127,194,190),(128,195,191),(133,201,141),(134,202,142),(135,203,143),(136,204,144),(149,170,210),(150,171,211),(151,172,212),(152,169,209),(153,175,213),(154,176,214),(155,173,215),(156,174,216)], [(1,49,29),(2,50,30),(3,51,31),(4,52,32),(5,48,108),(6,45,105),(7,46,106),(8,47,107),(9,43,57),(10,44,58),(11,41,59),(12,42,60),(13,137,34),(14,138,35),(15,139,36),(16,140,33),(17,61,37),(18,62,38),(19,63,39),(20,64,40),(21,165,86),(22,166,87),(23,167,88),(24,168,85),(25,117,211),(26,118,212),(27,119,209),(28,120,210),(53,128,158),(54,125,159),(55,126,160),(56,127,157),(65,95,78),(66,96,79),(67,93,80),(68,94,77),(69,99,112),(70,100,109),(71,97,110),(72,98,111),(73,102,89),(74,103,90),(75,104,91),(76,101,92),(81,173,202),(82,174,203),(83,175,204),(84,176,201),(113,146,190),(114,147,191),(115,148,192),(116,145,189),(121,155,134),(122,156,135),(123,153,136),(124,154,133),(129,162,149),(130,163,150),(131,164,151),(132,161,152),(141,185,214),(142,186,215),(143,187,216),(144,188,213),(169,198,182),(170,199,183),(171,200,184),(172,197,181),(177,207,194),(178,208,195),(179,205,196),(180,206,193)], [(1,80,43),(2,77,44),(3,78,41),(4,79,42),(5,104,17),(6,101,18),(7,102,19),(8,103,20),(9,29,93),(10,30,94),(11,31,95),(12,32,96),(13,85,110),(14,86,111),(15,87,112),(16,88,109),(21,72,138),(22,69,139),(23,70,140),(24,71,137),(25,157,175),(26,158,176),(27,159,173),(28,160,174),(33,167,100),(34,168,97),(35,165,98),(36,166,99),(37,108,75),(38,105,76),(39,106,73),(40,107,74),(45,92,62),(46,89,63),(47,90,64),(48,91,61),(49,67,57),(50,68,58),(51,65,59),(52,66,60),(53,201,118),(54,202,119),(55,203,120),(56,204,117),(81,209,125),(82,210,126),(83,211,127),(84,212,128),(113,153,200),(114,154,197),(115,155,198),(116,156,199),(121,169,192),(122,170,189),(123,171,190),(124,172,191),(129,180,216),(130,177,213),(131,178,214),(132,179,215),(133,181,147),(134,182,148),(135,183,145),(136,184,146),(141,164,208),(142,161,205),(143,162,206),(144,163,207),(149,193,187),(150,194,188),(151,195,185),(152,196,186)], [(1,148),(2,145),(3,146),(4,147),(5,120),(6,117),(7,118),(8,119),(9,121),(10,122),(11,123),(12,124),(13,161),(14,162),(15,163),(16,164),(17,55),(18,56),(19,53),(20,54),(21,216),(22,213),(23,214),(24,215),(25,45),(26,46),(27,47),(28,48),(29,192),(30,189),(31,190),(32,191),(33,151),(34,152),(35,149),(36,150),(37,126),(38,127),(39,128),(40,125),(41,136),(42,133),(43,134),(44,135),(49,115),(50,116),(51,113),(52,114),(57,155),(58,156),(59,153),(60,154),(61,160),(62,157),(63,158),(64,159),(65,200),(66,197),(67,198),(68,199),(69,177),(70,178),(71,179),(72,180),(73,84),(74,81),(75,82),(76,83),(77,183),(78,184),(79,181),(80,182),(85,142),(86,143),(87,144),(88,141),(89,176),(90,173),(91,174),(92,175),(93,169),(94,170),(95,171),(96,172),(97,196),(98,193),(99,194),(100,195),(101,204),(102,201),(103,202),(104,203),(105,211),(106,212),(107,209),(108,210),(109,208),(110,205),(111,206),(112,207),(129,138),(130,139),(131,140),(132,137),(165,187),(166,188),(167,185),(168,186)]])

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3M 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6AM 12A ··· 12AZ order 1 2 2 2 2 2 2 2 3 ··· 3 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 27 27 27 27 2 ··· 2 1 1 1 1 27 27 27 27 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4×S3 kernel C2×C4×C33⋊C2 C4×C33⋊C2 C2×C33⋊5C4 C3×C6×C12 C22×C33⋊C2 C2×C33⋊C2 C6×C12 C3×C12 C62 C3×C6 # reps 1 4 1 1 1 8 13 26 13 52

Matrix representation of C2×C4×C33⋊C2 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1] >;

C2×C4×C33⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3^3\rtimes C_2
% in TeX

G:=Group("C2xC4xC3^3:C2");
// GroupNames label

G:=SmallGroup(432,721);
// by ID

G=gap.SmallGroup(432,721);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,1124,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=c^3=d^3=e^3=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,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f=c^-1,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

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