Copied to
clipboard

## G = C33⋊15D8order 432 = 24·33

### 3rd semidirect product of C33 and D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12 — C33⋊15D8
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C33⋊12D4 — C33⋊15D8
 Lower central C33 — C32×C6 — C32×C12 — C33⋊15D8
 Upper central C1 — C2 — C4 — D4

Generators and relations for C3315D8
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=d-1 >

Subgroups: 1936 in 308 conjugacy classes, 115 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×13], C4, C22 [×2], S3 [×13], C6 [×13], C6 [×13], C8, D4, D4, C32 [×13], C12 [×13], D6 [×13], C2×C6 [×13], D8, C3⋊S3 [×13], C3×C6 [×13], C3×C6 [×13], C3⋊C8 [×13], D12 [×13], C3×D4 [×13], C33, C3×C12 [×13], C2×C3⋊S3 [×13], C62 [×13], D4⋊S3 [×13], C33⋊C2, C32×C6, C32×C6, C324C8 [×13], C12⋊S3 [×13], D4×C32 [×13], C32×C12, C2×C33⋊C2, C3×C62, C327D8 [×13], C337C8, C3312D4, D4×C33, C3315D8
Quotients: C1, C2 [×3], C22, S3 [×13], D4, D6 [×13], D8, C3⋊S3 [×13], C3⋊D4 [×13], C2×C3⋊S3 [×13], D4⋊S3 [×13], C33⋊C2, C327D4 [×13], C2×C33⋊C2, C327D8 [×13], C3315D4, C3315D8

Smallest permutation representation of C3315D8
On 216 points
Generators in S216
(1 162 152)(2 145 163)(3 164 146)(4 147 165)(5 166 148)(6 149 167)(7 168 150)(8 151 161)(9 183 31)(10 32 184)(11 177 25)(12 26 178)(13 179 27)(14 28 180)(15 181 29)(16 30 182)(17 108 173)(18 174 109)(19 110 175)(20 176 111)(21 112 169)(22 170 105)(23 106 171)(24 172 107)(33 88 127)(34 128 81)(35 82 121)(36 122 83)(37 84 123)(38 124 85)(39 86 125)(40 126 87)(41 133 195)(42 196 134)(43 135 197)(44 198 136)(45 129 199)(46 200 130)(47 131 193)(48 194 132)(49 158 120)(50 113 159)(51 160 114)(52 115 153)(53 154 116)(54 117 155)(55 156 118)(56 119 157)(57 67 201)(58 202 68)(59 69 203)(60 204 70)(61 71 205)(62 206 72)(63 65 207)(64 208 66)(73 93 192)(74 185 94)(75 95 186)(76 187 96)(77 89 188)(78 189 90)(79 91 190)(80 191 92)(97 215 140)(98 141 216)(99 209 142)(100 143 210)(101 211 144)(102 137 212)(103 213 138)(104 139 214)
(1 103 47)(2 48 104)(3 97 41)(4 42 98)(5 99 43)(6 44 100)(7 101 45)(8 46 102)(9 34 189)(10 190 35)(11 36 191)(12 192 37)(13 38 185)(14 186 39)(15 40 187)(16 188 33)(17 159 67)(18 68 160)(19 153 69)(20 70 154)(21 155 71)(22 72 156)(23 157 65)(24 66 158)(25 83 80)(26 73 84)(27 85 74)(28 75 86)(29 87 76)(30 77 88)(31 81 78)(32 79 82)(49 107 208)(50 201 108)(51 109 202)(52 203 110)(53 111 204)(54 205 112)(55 105 206)(56 207 106)(57 173 113)(58 114 174)(59 175 115)(60 116 176)(61 169 117)(62 118 170)(63 171 119)(64 120 172)(89 127 182)(90 183 128)(91 121 184)(92 177 122)(93 123 178)(94 179 124)(95 125 180)(96 181 126)(129 168 211)(130 212 161)(131 162 213)(132 214 163)(133 164 215)(134 216 165)(135 166 209)(136 210 167)(137 151 200)(138 193 152)(139 145 194)(140 195 146)(141 147 196)(142 197 148)(143 149 198)(144 199 150)
(1 81 117)(2 118 82)(3 83 119)(4 120 84)(5 85 113)(6 114 86)(7 87 115)(8 116 88)(9 21 131)(10 132 22)(11 23 133)(12 134 24)(13 17 135)(14 136 18)(15 19 129)(16 130 20)(25 171 41)(26 42 172)(27 173 43)(28 44 174)(29 175 45)(30 46 176)(31 169 47)(32 48 170)(33 161 154)(34 155 162)(35 163 156)(36 157 164)(37 165 158)(38 159 166)(39 167 160)(40 153 168)(49 123 147)(50 148 124)(51 125 149)(52 150 126)(53 127 151)(54 152 128)(55 121 145)(56 146 122)(57 99 74)(58 75 100)(59 101 76)(60 77 102)(61 103 78)(62 79 104)(63 97 80)(64 73 98)(65 215 191)(66 192 216)(67 209 185)(68 186 210)(69 211 187)(70 188 212)(71 213 189)(72 190 214)(89 137 204)(90 205 138)(91 139 206)(92 207 140)(93 141 208)(94 201 142)(95 143 202)(96 203 144)(105 184 194)(106 195 177)(107 178 196)(108 197 179)(109 180 198)(110 199 181)(111 182 200)(112 193 183)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 28)(26 27)(29 32)(30 31)(33 34)(35 40)(36 39)(37 38)(41 44)(42 43)(45 48)(46 47)(49 50)(51 56)(52 55)(53 54)(57 64)(58 63)(59 62)(60 61)(65 68)(66 67)(69 72)(70 71)(73 74)(75 80)(76 79)(77 78)(81 88)(82 87)(83 86)(84 85)(89 90)(91 96)(92 95)(93 94)(97 100)(98 99)(101 104)(102 103)(105 110)(106 109)(107 108)(111 112)(113 120)(114 119)(115 118)(116 117)(121 126)(122 125)(123 124)(127 128)(129 132)(130 131)(133 136)(134 135)(137 138)(139 144)(140 143)(141 142)(145 150)(146 149)(147 148)(151 152)(153 156)(154 155)(157 160)(158 159)(161 162)(163 168)(164 167)(165 166)(169 176)(170 175)(171 174)(172 173)(177 180)(178 179)(181 184)(182 183)(185 192)(186 191)(187 190)(188 189)(193 200)(194 199)(195 198)(196 197)(201 208)(202 207)(203 206)(204 205)(209 216)(210 215)(211 214)(212 213)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3M 4 6A ··· 6M 6N ··· 6AM 8A 8B 12A ··· 12M order 1 2 2 2 3 ··· 3 4 6 ··· 6 6 ··· 6 8 8 12 ··· 12 size 1 1 4 108 2 ··· 2 2 2 ··· 2 4 ··· 4 54 54 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D8 C3⋊D4 D4⋊S3 kernel C33⋊15D8 C33⋊7C8 C33⋊12D4 D4×C33 D4×C32 C32×C6 C3×C12 C33 C3×C6 C32 # reps 1 1 1 1 13 1 13 2 26 13

Matrix representation of C3315D8 in GL8(𝔽73)

 8 17 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 71 10 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 10 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 10 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 72 71 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 45 11 0 0 0 0 0 0 35 28 0 0 0 0 0 0 0 0 11 47 0 0 0 0 0 0 72 62 0 0 0 0 0 0 0 0 57 16 0 0 0 0 0 0 57 57
,
 1 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 45 11 0 0 0 0 0 0 35 28 0 0 0 0 0 0 0 0 11 47 0 0 0 0 0 0 72 62 0 0 0 0 0 0 0 0 57 16 0 0 0 0 0 0 16 16

G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,17,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,7,0,0,0,0,0,0,10,1,0,0,0,0,0,0,0,0,8,10,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,10,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,1,0,0,0,0,0,0,71,1,0,0,0,0,0,0,0,0,45,35,0,0,0,0,0,0,11,28,0,0,0,0,0,0,0,0,11,72,0,0,0,0,0,0,47,62,0,0,0,0,0,0,0,0,57,57,0,0,0,0,0,0,16,57],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,45,35,0,0,0,0,0,0,11,28,0,0,0,0,0,0,0,0,11,72,0,0,0,0,0,0,47,62,0,0,0,0,0,0,0,0,57,16,0,0,0,0,0,0,16,16] >;

C3315D8 in GAP, Magma, Sage, TeX

C_3^3\rtimes_{15}D_8
% in TeX

G:=Group("C3^3:15D8");
// GroupNames label

G:=SmallGroup(432,507);
// by ID

G=gap.SmallGroup(432,507);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,1124,4037,14118]);
// 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^-1>;
// generators/relations

׿
×
𝔽