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## G = D4×C32×C6order 432 = 24·33

### Direct product of C32×C6 and D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C32×C6
 Chief series C1 — C2 — C6 — C3×C6 — C32×C6 — C3×C62 — D4×C33 — D4×C32×C6
 Lower central C1 — C2 — D4×C32×C6
 Upper central C1 — C3×C62 — D4×C32×C6

Generators and relations for D4×C32×C6
G = < a,b,c,d,e | a3=b3=c6=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 980 in 756 conjugacy classes, 532 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, D4, C23, C32, C12, C2×C6, C2×C6, C2×D4, C3×C6, C3×C6, C2×C12, C3×D4, C22×C6, C33, C3×C12, C62, C62, C6×D4, C32×C6, C32×C6, C32×C6, C6×C12, D4×C32, C2×C62, C32×C12, C3×C62, C3×C62, C3×C62, D4×C3×C6, C3×C6×C12, D4×C33, C63, D4×C32×C6
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C3×C6, C3×D4, C22×C6, C33, C62, C6×D4, C32×C6, D4×C32, C2×C62, C3×C62, D4×C3×C6, D4×C33, C63, D4×C32×C6

Smallest permutation representation of D4×C32×C6
On 216 points
Generators in S216
(1 112 48)(2 113 43)(3 114 44)(4 109 45)(5 110 46)(6 111 47)(7 33 23)(8 34 24)(9 35 19)(10 36 20)(11 31 21)(12 32 22)(13 84 68)(14 79 69)(15 80 70)(16 81 71)(17 82 72)(18 83 67)(25 78 62)(26 73 63)(27 74 64)(28 75 65)(29 76 66)(30 77 61)(37 104 94)(38 105 95)(39 106 96)(40 107 91)(41 108 92)(42 103 93)(49 98 88)(50 99 89)(51 100 90)(52 101 85)(53 102 86)(54 97 87)(55 156 140)(56 151 141)(57 152 142)(58 153 143)(59 154 144)(60 155 139)(115 182 131)(116 183 132)(117 184 127)(118 185 128)(119 186 129)(120 181 130)(121 170 137)(122 171 138)(123 172 133)(124 173 134)(125 174 135)(126 169 136)(145 212 202)(146 213 203)(147 214 204)(148 215 199)(149 216 200)(150 211 201)(157 206 196)(158 207 197)(159 208 198)(160 209 193)(161 210 194)(162 205 195)(163 189 179)(164 190 180)(165 191 175)(166 192 176)(167 187 177)(168 188 178)
(1 38 85)(2 39 86)(3 40 87)(4 41 88)(5 42 89)(6 37 90)(7 13 64)(8 14 65)(9 15 66)(10 16 61)(11 17 62)(12 18 63)(19 70 76)(20 71 77)(21 72 78)(22 67 73)(23 68 74)(24 69 75)(25 31 82)(26 32 83)(27 33 84)(28 34 79)(29 35 80)(30 36 81)(43 96 102)(44 91 97)(45 92 98)(46 93 99)(47 94 100)(48 95 101)(49 109 108)(50 110 103)(51 111 104)(52 112 105)(53 113 106)(54 114 107)(55 213 160)(56 214 161)(57 215 162)(58 216 157)(59 211 158)(60 212 159)(115 168 174)(116 163 169)(117 164 170)(118 165 171)(119 166 172)(120 167 173)(121 127 180)(122 128 175)(123 129 176)(124 130 177)(125 131 178)(126 132 179)(133 186 192)(134 181 187)(135 182 188)(136 183 189)(137 184 190)(138 185 191)(139 145 198)(140 146 193)(141 147 194)(142 148 195)(143 149 196)(144 150 197)(151 204 210)(152 199 205)(153 200 206)(154 201 207)(155 202 208)(156 203 209)
(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)
(1 143 10 163)(2 144 11 164)(3 139 12 165)(4 140 7 166)(5 141 8 167)(6 142 9 168)(13 172 41 146)(14 173 42 147)(15 174 37 148)(16 169 38 149)(17 170 39 150)(18 171 40 145)(19 178 47 152)(20 179 48 153)(21 180 43 154)(22 175 44 155)(23 176 45 156)(24 177 46 151)(25 184 53 158)(26 185 54 159)(27 186 49 160)(28 181 50 161)(29 182 51 162)(30 183 52 157)(31 190 113 59)(32 191 114 60)(33 192 109 55)(34 187 110 56)(35 188 111 57)(36 189 112 58)(61 116 85 196)(62 117 86 197)(63 118 87 198)(64 119 88 193)(65 120 89 194)(66 115 90 195)(67 122 91 202)(68 123 92 203)(69 124 93 204)(70 125 94 199)(71 126 95 200)(72 121 96 201)(73 128 97 208)(74 129 98 209)(75 130 99 210)(76 131 100 205)(77 132 101 206)(78 127 102 207)(79 134 103 214)(80 135 104 215)(81 136 105 216)(82 137 106 211)(83 138 107 212)(84 133 108 213)
(1 163)(2 164)(3 165)(4 166)(5 167)(6 168)(7 140)(8 141)(9 142)(10 143)(11 144)(12 139)(13 146)(14 147)(15 148)(16 149)(17 150)(18 145)(19 152)(20 153)(21 154)(22 155)(23 156)(24 151)(25 158)(26 159)(27 160)(28 161)(29 162)(30 157)(31 59)(32 60)(33 55)(34 56)(35 57)(36 58)(37 174)(38 169)(39 170)(40 171)(41 172)(42 173)(43 180)(44 175)(45 176)(46 177)(47 178)(48 179)(49 186)(50 181)(51 182)(52 183)(53 184)(54 185)(61 196)(62 197)(63 198)(64 193)(65 194)(66 195)(67 202)(68 203)(69 204)(70 199)(71 200)(72 201)(73 208)(74 209)(75 210)(76 205)(77 206)(78 207)(79 214)(80 215)(81 216)(82 211)(83 212)(84 213)(85 116)(86 117)(87 118)(88 119)(89 120)(90 115)(91 122)(92 123)(93 124)(94 125)(95 126)(96 121)(97 128)(98 129)(99 130)(100 131)(101 132)(102 127)(103 134)(104 135)(105 136)(106 137)(107 138)(108 133)(109 192)(110 187)(111 188)(112 189)(113 190)(114 191)

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

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

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

270 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3Z 4A 4B 6A ··· 6BZ 6CA ··· 6FZ 12A ··· 12AZ order 1 2 2 2 2 2 2 2 3 ··· 3 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 1 ··· 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2

270 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 kernel D4×C32×C6 C3×C6×C12 D4×C33 C63 D4×C3×C6 C6×C12 D4×C32 C2×C62 C32×C6 C3×C6 # reps 1 1 4 2 26 26 104 52 2 52

Matrix representation of D4×C32×C6 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 12 0 0 0 0 4 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 1 0
,
 12 0 0 0 0 12 0 0 0 0 0 12 0 0 12 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,4,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[12,0,0,0,0,12,0,0,0,0,0,12,0,0,12,0] >;

D4×C32×C6 in GAP, Magma, Sage, TeX

D_4\times C_3^2\times C_6
% in TeX

G:=Group("D4xC3^2xC6");
// GroupNames label

G:=SmallGroup(432,731);
// by ID

G=gap.SmallGroup(432,731);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-3,-2,3053]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^6=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽