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G = C33⋊12D8order 432 = 24·33

3rd semidirect product of C33 and D8 acting via D8/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12 — C33⋊12D8
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C33⋊12D4 — C33⋊12D8
 Lower central C33 — C32×C6 — C32×C12 — C33⋊12D8
 Upper central C1 — C2 — C4 — C8

Generators and relations for C3312D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 2872 in 308 conjugacy classes, 115 normal (9 characteristic)
C1, C2, C2 [×2], C3 [×13], C4, C22 [×2], S3 [×26], C6 [×13], C8, D4 [×2], C32 [×13], C12 [×13], D6 [×26], D8, C3⋊S3 [×26], C3×C6 [×13], C24 [×13], D12 [×26], C33, C3×C12 [×13], C2×C3⋊S3 [×26], D24 [×13], C33⋊C2 [×2], C32×C6, C3×C24 [×13], C12⋊S3 [×26], C32×C12, C2×C33⋊C2 [×2], C325D8 [×13], C32×C24, C3312D4 [×2], C3312D8
Quotients: C1, C2 [×3], C22, S3 [×13], D4, D6 [×13], D8, C3⋊S3 [×13], D12 [×13], C2×C3⋊S3 [×13], D24 [×13], C33⋊C2, C12⋊S3 [×13], C2×C33⋊C2, C325D8 [×13], C3312D4, C3312D8

Smallest permutation representation of C3312D8
On 216 points
Generators in S216
```(1 31 117)(2 32 118)(3 25 119)(4 26 120)(5 27 113)(6 28 114)(7 29 115)(8 30 116)(9 182 187)(10 183 188)(11 184 189)(12 177 190)(13 178 191)(14 179 192)(15 180 185)(16 181 186)(17 132 73)(18 133 74)(19 134 75)(20 135 76)(21 136 77)(22 129 78)(23 130 79)(24 131 80)(33 155 88)(34 156 81)(35 157 82)(36 158 83)(37 159 84)(38 160 85)(39 153 86)(40 154 87)(41 109 149)(42 110 150)(43 111 151)(44 112 152)(45 105 145)(46 106 146)(47 107 147)(48 108 148)(49 70 127)(50 71 128)(51 72 121)(52 65 122)(53 66 123)(54 67 124)(55 68 125)(56 69 126)(57 96 203)(58 89 204)(59 90 205)(60 91 206)(61 92 207)(62 93 208)(63 94 201)(64 95 202)(97 166 171)(98 167 172)(99 168 173)(100 161 174)(101 162 175)(102 163 176)(103 164 169)(104 165 170)(137 193 212)(138 194 213)(139 195 214)(140 196 215)(141 197 216)(142 198 209)(143 199 210)(144 200 211)
(1 194 20)(2 195 21)(3 196 22)(4 197 23)(5 198 24)(6 199 17)(7 200 18)(8 193 19)(9 168 150)(10 161 151)(11 162 152)(12 163 145)(13 164 146)(14 165 147)(15 166 148)(16 167 149)(25 215 129)(26 216 130)(27 209 131)(28 210 132)(29 211 133)(30 212 134)(31 213 135)(32 214 136)(33 70 58)(34 71 59)(35 72 60)(36 65 61)(37 66 62)(38 67 63)(39 68 64)(40 69 57)(41 181 172)(42 182 173)(43 183 174)(44 184 175)(45 177 176)(46 178 169)(47 179 170)(48 180 171)(49 204 88)(50 205 81)(51 206 82)(52 207 83)(53 208 84)(54 201 85)(55 202 86)(56 203 87)(73 114 143)(74 115 144)(75 116 137)(76 117 138)(77 118 139)(78 119 140)(79 120 141)(80 113 142)(89 155 127)(90 156 128)(91 157 121)(92 158 122)(93 159 123)(94 160 124)(95 153 125)(96 154 126)(97 108 185)(98 109 186)(99 110 187)(100 111 188)(101 112 189)(102 105 190)(103 106 191)(104 107 192)
(1 185 71)(2 186 72)(3 187 65)(4 188 66)(5 189 67)(6 190 68)(7 191 69)(8 192 70)(9 122 25)(10 123 26)(11 124 27)(12 125 28)(13 126 29)(14 127 30)(15 128 31)(16 121 32)(17 105 39)(18 106 40)(19 107 33)(20 108 34)(21 109 35)(22 110 36)(23 111 37)(24 112 38)(41 82 77)(42 83 78)(43 84 79)(44 85 80)(45 86 73)(46 87 74)(47 88 75)(48 81 76)(49 116 179)(50 117 180)(51 118 181)(52 119 182)(53 120 183)(54 113 184)(55 114 177)(56 115 178)(57 200 103)(58 193 104)(59 194 97)(60 195 98)(61 196 99)(62 197 100)(63 198 101)(64 199 102)(89 212 165)(90 213 166)(91 214 167)(92 215 168)(93 216 161)(94 209 162)(95 210 163)(96 211 164)(129 150 158)(130 151 159)(131 152 160)(132 145 153)(133 146 154)(134 147 155)(135 148 156)(136 149 157)(137 170 204)(138 171 205)(139 172 206)(140 173 207)(141 174 208)(142 175 201)(143 176 202)(144 169 203)
(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 7)(2 6)(3 5)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 56)(16 55)(17 195)(18 194)(19 193)(20 200)(21 199)(22 198)(23 197)(24 196)(25 113)(26 120)(27 119)(28 118)(29 117)(30 116)(31 115)(32 114)(33 104)(34 103)(35 102)(36 101)(37 100)(38 99)(39 98)(40 97)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 96)(57 108)(58 107)(59 106)(60 105)(61 112)(62 111)(63 110)(64 109)(65 189)(66 188)(67 187)(68 186)(69 185)(70 192)(71 191)(72 190)(73 214)(74 213)(75 212)(76 211)(77 210)(78 209)(79 216)(80 215)(81 164)(82 163)(83 162)(84 161)(85 168)(86 167)(87 166)(88 165)(121 177)(122 184)(123 183)(124 182)(125 181)(126 180)(127 179)(128 178)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 144)(136 143)(145 206)(146 205)(147 204)(148 203)(149 202)(150 201)(151 208)(152 207)(153 172)(154 171)(155 170)(156 169)(157 176)(158 175)(159 174)(160 173)```

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

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

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

111 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3M 4 6A ··· 6M 8A 8B 12A ··· 12Z 24A ··· 24AZ order 1 2 2 2 3 ··· 3 4 6 ··· 6 8 8 12 ··· 12 24 ··· 24 size 1 1 108 108 2 ··· 2 2 2 ··· 2 2 2 2 ··· 2 2 ··· 2

111 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D4 D6 D8 D12 D24 kernel C33⋊12D8 C32×C24 C33⋊12D4 C3×C24 C32×C6 C3×C12 C33 C3×C6 C32 # reps 1 1 2 13 1 13 2 26 52

Matrix representation of C3312D8 in GL6(𝔽73)

 0 1 0 0 0 0 72 72 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 72 72 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 18 68 0 0 0 0 5 23 0 0 0 0 0 0 55 5 0 0 0 0 68 50 0 0 0 0 0 0 18 68 0 0 0 0 5 23
,
 14 7 0 0 0 0 66 59 0 0 0 0 0 0 7 14 0 0 0 0 7 66 0 0 0 0 0 0 7 14 0 0 0 0 7 66

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[18,5,0,0,0,0,68,23,0,0,0,0,0,0,55,68,0,0,0,0,5,50,0,0,0,0,0,0,18,5,0,0,0,0,68,23],[14,66,0,0,0,0,7,59,0,0,0,0,0,0,7,7,0,0,0,0,14,66,0,0,0,0,0,0,7,7,0,0,0,0,14,66] >;`

C3312D8 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_{12}D_8`
`% in TeX`

`G:=Group("C3^3:12D8");`
`// GroupNames label`

`G:=SmallGroup(432,499);`
`// by ID`

`G=gap.SmallGroup(432,499);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,1124,4037,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;`
`// generators/relations`

׿
×
𝔽