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## G = C33⋊15M4(2)  order 432 = 24·33

### 3rd semidirect product of C33 and M4(2) acting via M4(2)/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊15M4(2)
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C4×C33⋊C2 — C33⋊15M4(2)
 Lower central C33 — C32×C6 — C33⋊15M4(2)
 Upper central C1 — C4 — C8

Generators and relations for C3315M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d5 >

Subgroups: 1400 in 280 conjugacy classes, 117 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, M4(2), C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C33, C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3, C33⋊C2, C32×C6, C324C8, C3×C24, C4×C3⋊S3, C335C4, C32×C12, C2×C33⋊C2, C24⋊S3, C337C8, C32×C24, C4×C33⋊C2, C3315M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, M4(2), C3⋊S3, C4×S3, C2×C3⋊S3, C8⋊S3, C33⋊C2, C4×C3⋊S3, C2×C33⋊C2, C24⋊S3, C4×C33⋊C2, C3315M4(2)

Smallest permutation representation of C3315M4(2)
On 216 points
Generators in S216
```(1 172 131)(2 132 173)(3 174 133)(4 134 175)(5 176 135)(6 136 169)(7 170 129)(8 130 171)(9 65 164)(10 165 66)(11 67 166)(12 167 68)(13 69 168)(14 161 70)(15 71 162)(16 163 72)(17 157 215)(18 216 158)(19 159 209)(20 210 160)(21 153 211)(22 212 154)(23 155 213)(24 214 156)(25 203 147)(26 148 204)(27 205 149)(28 150 206)(29 207 151)(30 152 208)(31 201 145)(32 146 202)(33 142 106)(34 107 143)(35 144 108)(36 109 137)(37 138 110)(38 111 139)(39 140 112)(40 105 141)(41 49 116)(42 117 50)(43 51 118)(44 119 52)(45 53 120)(46 113 54)(47 55 114)(48 115 56)(57 94 178)(58 179 95)(59 96 180)(60 181 89)(61 90 182)(62 183 91)(63 92 184)(64 177 93)(73 125 86)(74 87 126)(75 127 88)(76 81 128)(77 121 82)(78 83 122)(79 123 84)(80 85 124)(97 189 198)(98 199 190)(99 191 200)(100 193 192)(101 185 194)(102 195 186)(103 187 196)(104 197 188)
(1 128 23)(2 24 121)(3 122 17)(4 18 123)(5 124 19)(6 20 125)(7 126 21)(8 22 127)(9 29 106)(10 107 30)(11 31 108)(12 109 32)(13 25 110)(14 111 26)(15 27 112)(16 105 28)(33 65 207)(34 208 66)(35 67 201)(36 202 68)(37 69 203)(38 204 70)(39 71 205)(40 206 72)(41 183 197)(42 198 184)(43 177 199)(44 200 178)(45 179 193)(46 194 180)(47 181 195)(48 196 182)(49 91 188)(50 189 92)(51 93 190)(52 191 94)(53 95 192)(54 185 96)(55 89 186)(56 187 90)(57 119 99)(58 100 120)(59 113 101)(60 102 114)(61 115 103)(62 104 116)(63 117 97)(64 98 118)(73 169 160)(74 153 170)(75 171 154)(76 155 172)(77 173 156)(78 157 174)(79 175 158)(80 159 176)(81 213 131)(82 132 214)(83 215 133)(84 134 216)(85 209 135)(86 136 210)(87 211 129)(88 130 212)(137 146 167)(138 168 147)(139 148 161)(140 162 149)(141 150 163)(142 164 151)(143 152 165)(144 166 145)
(1 117 31)(2 32 118)(3 119 25)(4 26 120)(5 113 27)(6 28 114)(7 115 29)(8 30 116)(9 21 61)(10 62 22)(11 23 63)(12 64 24)(13 17 57)(14 58 18)(15 19 59)(16 60 20)(33 74 187)(34 188 75)(35 76 189)(36 190 77)(37 78 191)(38 192 79)(39 80 185)(40 186 73)(41 130 152)(42 145 131)(43 132 146)(44 147 133)(45 134 148)(46 149 135)(47 136 150)(48 151 129)(49 171 208)(50 201 172)(51 173 202)(52 203 174)(53 175 204)(54 205 176)(55 169 206)(56 207 170)(65 153 90)(66 91 154)(67 155 92)(68 93 156)(69 157 94)(70 95 158)(71 159 96)(72 89 160)(81 198 144)(82 137 199)(83 200 138)(84 139 193)(85 194 140)(86 141 195)(87 196 142)(88 143 197)(97 108 128)(98 121 109)(99 110 122)(100 123 111)(101 112 124)(102 125 105)(103 106 126)(104 127 107)(161 179 216)(162 209 180)(163 181 210)(164 211 182)(165 183 212)(166 213 184)(167 177 214)(168 215 178)
(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)
(2 6)(4 8)(9 103)(10 100)(11 97)(12 102)(13 99)(14 104)(15 101)(16 98)(17 122)(18 127)(19 124)(20 121)(21 126)(22 123)(23 128)(24 125)(25 119)(26 116)(27 113)(28 118)(29 115)(30 120)(31 117)(32 114)(33 182)(34 179)(35 184)(36 181)(37 178)(38 183)(39 180)(40 177)(41 204)(42 201)(43 206)(44 203)(45 208)(46 205)(47 202)(48 207)(49 148)(50 145)(51 150)(52 147)(53 152)(54 149)(55 146)(56 151)(57 110)(58 107)(59 112)(60 109)(61 106)(62 111)(63 108)(64 105)(65 196)(66 193)(67 198)(68 195)(69 200)(70 197)(71 194)(72 199)(73 214)(74 211)(75 216)(76 213)(77 210)(78 215)(79 212)(80 209)(81 155)(82 160)(83 157)(84 154)(85 159)(86 156)(87 153)(88 158)(89 137)(90 142)(91 139)(92 144)(93 141)(94 138)(95 143)(96 140)(129 170)(130 175)(131 172)(132 169)(133 174)(134 171)(135 176)(136 173)(161 188)(162 185)(163 190)(164 187)(165 192)(166 189)(167 186)(168 191)```

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

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

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

114 conjugacy classes

 class 1 2A 2B 3A ··· 3M 4A 4B 4C 6A ··· 6M 8A 8B 8C 8D 12A ··· 12Z 24A ··· 24AZ order 1 2 2 3 ··· 3 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 54 2 ··· 2 1 1 54 2 ··· 2 2 2 54 54 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 S3 D6 M4(2) C4×S3 C8⋊S3 kernel C33⋊15M4(2) C33⋊7C8 C32×C24 C4×C33⋊C2 C33⋊5C4 C2×C33⋊C2 C3×C24 C3×C12 C33 C3×C6 C32 # reps 1 1 1 1 2 2 13 13 2 26 52

Matrix representation of C3315M4(2) in GL6(𝔽73)

 1 0 0 0 0 0 0 1 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
,
 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 67 3 0 0 0 0 70 6 0 0 0 0 0 0 70 6 0 0 0 0 3 3
,
 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 1 0 0 0 0 0 1 72

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,67,70,0,0,0,0,3,6,0,0,0,0,0,0,70,3,0,0,0,0,6,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;`

C3315M4(2) in GAP, Magma, Sage, TeX

`C_3^3\rtimes_{15}M_4(2)`
`% in TeX`

`G:=Group("C3^3:15M4(2)");`
`// GroupNames label`

`G:=SmallGroup(432,497);`
`// by ID`

`G=gap.SmallGroup(432,497);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,1124,4037,14118]);`
`// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;`
`// generators/relations`

׿
×
𝔽