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## G = C4⋊C4⋊D15order 480 = 25·3·5

### 6th semidirect product of C4⋊C4 and D15 acting via D15/C15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C4⋊C4⋊D15
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — D30⋊3C4 — C4⋊C4⋊D15
 Lower central C15 — C2×C30 — C4⋊C4⋊D15
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊C4⋊D15
G = < a,b,c,d | a4=b4=c15=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 708 in 120 conjugacy classes, 47 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, S3, C6 [×3], C2×C4 [×3], C2×C4 [×3], C23, D5, C10 [×3], Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C15, C42, C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×2], Dic5 [×3], C20 [×3], D10 [×3], C2×C10, C2×Dic3 [×3], C2×C12 [×3], C22×S3, D15, C30 [×3], C422C2, C2×Dic5 [×3], C2×C20 [×3], C22×D5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C3×C4⋊C4, Dic15 [×3], C60 [×3], D30 [×3], C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4 [×3], C5×C4⋊C4, C4⋊C4⋊S3, C2×Dic15 [×3], C2×C60 [×3], C22×D15, C4⋊C4⋊D5, C4×Dic15, C30.4Q8, C605C4, D303C4 [×3], C15×C4⋊C4, C4⋊C4⋊D15
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, D15, C422C2, C22×D5, C4○D12, D42S3, Q83S3, D30 [×3], C4○D20, D42D5, Q82D5, C4⋊C4⋊S3, C22×D15, C4⋊C4⋊D5, D6011C2, D42D15, Q83D15, C4⋊C4⋊D15

Smallest permutation representation of C4⋊C4⋊D15
On 240 points
Generators in S240
```(1 118 20 103)(2 119 21 104)(3 120 22 105)(4 106 23 91)(5 107 24 92)(6 108 25 93)(7 109 26 94)(8 110 27 95)(9 111 28 96)(10 112 29 97)(11 113 30 98)(12 114 16 99)(13 115 17 100)(14 116 18 101)(15 117 19 102)(31 87 57 72)(32 88 58 73)(33 89 59 74)(34 90 60 75)(35 76 46 61)(36 77 47 62)(37 78 48 63)(38 79 49 64)(39 80 50 65)(40 81 51 66)(41 82 52 67)(42 83 53 68)(43 84 54 69)(44 85 55 70)(45 86 56 71)(121 211 139 229)(122 212 140 230)(123 213 141 231)(124 214 142 232)(125 215 143 233)(126 216 144 234)(127 217 145 235)(128 218 146 236)(129 219 147 237)(130 220 148 238)(131 221 149 239)(132 222 150 240)(133 223 136 226)(134 224 137 227)(135 225 138 228)(151 189 174 200)(152 190 175 201)(153 191 176 202)(154 192 177 203)(155 193 178 204)(156 194 179 205)(157 195 180 206)(158 181 166 207)(159 182 167 208)(160 183 168 209)(161 184 169 210)(162 185 170 196)(163 186 171 197)(164 187 172 198)(165 188 173 199)
(1 155 32 133)(2 156 33 134)(3 157 34 135)(4 158 35 121)(5 159 36 122)(6 160 37 123)(7 161 38 124)(8 162 39 125)(9 163 40 126)(10 164 41 127)(11 165 42 128)(12 151 43 129)(13 152 44 130)(14 153 45 131)(15 154 31 132)(16 174 54 147)(17 175 55 148)(18 176 56 149)(19 177 57 150)(20 178 58 136)(21 179 59 137)(22 180 60 138)(23 166 46 139)(24 167 47 140)(25 168 48 141)(26 169 49 142)(27 170 50 143)(28 171 51 144)(29 172 52 145)(30 173 53 146)(61 211 91 181)(62 212 92 182)(63 213 93 183)(64 214 94 184)(65 215 95 185)(66 216 96 186)(67 217 97 187)(68 218 98 188)(69 219 99 189)(70 220 100 190)(71 221 101 191)(72 222 102 192)(73 223 103 193)(74 224 104 194)(75 225 105 195)(76 229 106 207)(77 230 107 208)(78 231 108 209)(79 232 109 210)(80 233 110 196)(81 234 111 197)(82 235 112 198)(83 236 113 199)(84 237 114 200)(85 238 115 201)(86 239 116 202)(87 240 117 203)(88 226 118 204)(89 227 119 205)(90 228 120 206)
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)(226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 23)(17 22)(18 21)(19 20)(24 30)(25 29)(26 28)(31 32)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 114)(77 113)(78 112)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(121 147)(122 146)(123 145)(124 144)(125 143)(126 142)(127 141)(128 140)(129 139)(130 138)(131 137)(132 136)(133 150)(134 149)(135 148)(151 166)(152 180)(153 179)(154 178)(155 177)(156 176)(157 175)(158 174)(159 173)(160 172)(161 171)(162 170)(163 169)(164 168)(165 167)(181 237)(182 236)(183 235)(184 234)(185 233)(186 232)(187 231)(188 230)(189 229)(190 228)(191 227)(192 226)(193 240)(194 239)(195 238)(196 215)(197 214)(198 213)(199 212)(200 211)(201 225)(202 224)(203 223)(204 222)(205 221)(206 220)(207 219)(208 218)(209 217)(210 216)```

`G:=sub<Sym(240)| (1,118,20,103)(2,119,21,104)(3,120,22,105)(4,106,23,91)(5,107,24,92)(6,108,25,93)(7,109,26,94)(8,110,27,95)(9,111,28,96)(10,112,29,97)(11,113,30,98)(12,114,16,99)(13,115,17,100)(14,116,18,101)(15,117,19,102)(31,87,57,72)(32,88,58,73)(33,89,59,74)(34,90,60,75)(35,76,46,61)(36,77,47,62)(37,78,48,63)(38,79,49,64)(39,80,50,65)(40,81,51,66)(41,82,52,67)(42,83,53,68)(43,84,54,69)(44,85,55,70)(45,86,56,71)(121,211,139,229)(122,212,140,230)(123,213,141,231)(124,214,142,232)(125,215,143,233)(126,216,144,234)(127,217,145,235)(128,218,146,236)(129,219,147,237)(130,220,148,238)(131,221,149,239)(132,222,150,240)(133,223,136,226)(134,224,137,227)(135,225,138,228)(151,189,174,200)(152,190,175,201)(153,191,176,202)(154,192,177,203)(155,193,178,204)(156,194,179,205)(157,195,180,206)(158,181,166,207)(159,182,167,208)(160,183,168,209)(161,184,169,210)(162,185,170,196)(163,186,171,197)(164,187,172,198)(165,188,173,199), (1,155,32,133)(2,156,33,134)(3,157,34,135)(4,158,35,121)(5,159,36,122)(6,160,37,123)(7,161,38,124)(8,162,39,125)(9,163,40,126)(10,164,41,127)(11,165,42,128)(12,151,43,129)(13,152,44,130)(14,153,45,131)(15,154,31,132)(16,174,54,147)(17,175,55,148)(18,176,56,149)(19,177,57,150)(20,178,58,136)(21,179,59,137)(22,180,60,138)(23,166,46,139)(24,167,47,140)(25,168,48,141)(26,169,49,142)(27,170,50,143)(28,171,51,144)(29,172,52,145)(30,173,53,146)(61,211,91,181)(62,212,92,182)(63,213,93,183)(64,214,94,184)(65,215,95,185)(66,216,96,186)(67,217,97,187)(68,218,98,188)(69,219,99,189)(70,220,100,190)(71,221,101,191)(72,222,102,192)(73,223,103,193)(74,224,104,194)(75,225,105,195)(76,229,106,207)(77,230,107,208)(78,231,108,209)(79,232,109,210)(80,233,110,196)(81,234,111,197)(82,235,112,198)(83,236,113,199)(84,237,114,200)(85,238,115,201)(86,239,116,202)(87,240,117,203)(88,226,118,204)(89,227,119,205)(90,228,120,206), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,23)(17,22)(18,21)(19,20)(24,30)(25,29)(26,28)(31,32)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,150)(134,149)(135,148)(151,166)(152,180)(153,179)(154,178)(155,177)(156,176)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(181,237)(182,236)(183,235)(184,234)(185,233)(186,232)(187,231)(188,230)(189,229)(190,228)(191,227)(192,226)(193,240)(194,239)(195,238)(196,215)(197,214)(198,213)(199,212)(200,211)(201,225)(202,224)(203,223)(204,222)(205,221)(206,220)(207,219)(208,218)(209,217)(210,216)>;`

`G:=Group( (1,118,20,103)(2,119,21,104)(3,120,22,105)(4,106,23,91)(5,107,24,92)(6,108,25,93)(7,109,26,94)(8,110,27,95)(9,111,28,96)(10,112,29,97)(11,113,30,98)(12,114,16,99)(13,115,17,100)(14,116,18,101)(15,117,19,102)(31,87,57,72)(32,88,58,73)(33,89,59,74)(34,90,60,75)(35,76,46,61)(36,77,47,62)(37,78,48,63)(38,79,49,64)(39,80,50,65)(40,81,51,66)(41,82,52,67)(42,83,53,68)(43,84,54,69)(44,85,55,70)(45,86,56,71)(121,211,139,229)(122,212,140,230)(123,213,141,231)(124,214,142,232)(125,215,143,233)(126,216,144,234)(127,217,145,235)(128,218,146,236)(129,219,147,237)(130,220,148,238)(131,221,149,239)(132,222,150,240)(133,223,136,226)(134,224,137,227)(135,225,138,228)(151,189,174,200)(152,190,175,201)(153,191,176,202)(154,192,177,203)(155,193,178,204)(156,194,179,205)(157,195,180,206)(158,181,166,207)(159,182,167,208)(160,183,168,209)(161,184,169,210)(162,185,170,196)(163,186,171,197)(164,187,172,198)(165,188,173,199), (1,155,32,133)(2,156,33,134)(3,157,34,135)(4,158,35,121)(5,159,36,122)(6,160,37,123)(7,161,38,124)(8,162,39,125)(9,163,40,126)(10,164,41,127)(11,165,42,128)(12,151,43,129)(13,152,44,130)(14,153,45,131)(15,154,31,132)(16,174,54,147)(17,175,55,148)(18,176,56,149)(19,177,57,150)(20,178,58,136)(21,179,59,137)(22,180,60,138)(23,166,46,139)(24,167,47,140)(25,168,48,141)(26,169,49,142)(27,170,50,143)(28,171,51,144)(29,172,52,145)(30,173,53,146)(61,211,91,181)(62,212,92,182)(63,213,93,183)(64,214,94,184)(65,215,95,185)(66,216,96,186)(67,217,97,187)(68,218,98,188)(69,219,99,189)(70,220,100,190)(71,221,101,191)(72,222,102,192)(73,223,103,193)(74,224,104,194)(75,225,105,195)(76,229,106,207)(77,230,107,208)(78,231,108,209)(79,232,109,210)(80,233,110,196)(81,234,111,197)(82,235,112,198)(83,236,113,199)(84,237,114,200)(85,238,115,201)(86,239,116,202)(87,240,117,203)(88,226,118,204)(89,227,119,205)(90,228,120,206), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,23)(17,22)(18,21)(19,20)(24,30)(25,29)(26,28)(31,32)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,150)(134,149)(135,148)(151,166)(152,180)(153,179)(154,178)(155,177)(156,176)(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)(181,237)(182,236)(183,235)(184,234)(185,233)(186,232)(187,231)(188,230)(189,229)(190,228)(191,227)(192,226)(193,240)(194,239)(195,238)(196,215)(197,214)(198,213)(199,212)(200,211)(201,225)(202,224)(203,223)(204,222)(205,221)(206,220)(207,219)(208,218)(209,217)(210,216) );`

`G=PermutationGroup([(1,118,20,103),(2,119,21,104),(3,120,22,105),(4,106,23,91),(5,107,24,92),(6,108,25,93),(7,109,26,94),(8,110,27,95),(9,111,28,96),(10,112,29,97),(11,113,30,98),(12,114,16,99),(13,115,17,100),(14,116,18,101),(15,117,19,102),(31,87,57,72),(32,88,58,73),(33,89,59,74),(34,90,60,75),(35,76,46,61),(36,77,47,62),(37,78,48,63),(38,79,49,64),(39,80,50,65),(40,81,51,66),(41,82,52,67),(42,83,53,68),(43,84,54,69),(44,85,55,70),(45,86,56,71),(121,211,139,229),(122,212,140,230),(123,213,141,231),(124,214,142,232),(125,215,143,233),(126,216,144,234),(127,217,145,235),(128,218,146,236),(129,219,147,237),(130,220,148,238),(131,221,149,239),(132,222,150,240),(133,223,136,226),(134,224,137,227),(135,225,138,228),(151,189,174,200),(152,190,175,201),(153,191,176,202),(154,192,177,203),(155,193,178,204),(156,194,179,205),(157,195,180,206),(158,181,166,207),(159,182,167,208),(160,183,168,209),(161,184,169,210),(162,185,170,196),(163,186,171,197),(164,187,172,198),(165,188,173,199)], [(1,155,32,133),(2,156,33,134),(3,157,34,135),(4,158,35,121),(5,159,36,122),(6,160,37,123),(7,161,38,124),(8,162,39,125),(9,163,40,126),(10,164,41,127),(11,165,42,128),(12,151,43,129),(13,152,44,130),(14,153,45,131),(15,154,31,132),(16,174,54,147),(17,175,55,148),(18,176,56,149),(19,177,57,150),(20,178,58,136),(21,179,59,137),(22,180,60,138),(23,166,46,139),(24,167,47,140),(25,168,48,141),(26,169,49,142),(27,170,50,143),(28,171,51,144),(29,172,52,145),(30,173,53,146),(61,211,91,181),(62,212,92,182),(63,213,93,183),(64,214,94,184),(65,215,95,185),(66,216,96,186),(67,217,97,187),(68,218,98,188),(69,219,99,189),(70,220,100,190),(71,221,101,191),(72,222,102,192),(73,223,103,193),(74,224,104,194),(75,225,105,195),(76,229,106,207),(77,230,107,208),(78,231,108,209),(79,232,109,210),(80,233,110,196),(81,234,111,197),(82,235,112,198),(83,236,113,199),(84,237,114,200),(85,238,115,201),(86,239,116,202),(87,240,117,203),(88,226,118,204),(89,227,119,205),(90,228,120,206)], [(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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225),(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,23),(17,22),(18,21),(19,20),(24,30),(25,29),(26,28),(31,32),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(46,54),(47,53),(48,52),(49,51),(55,60),(56,59),(57,58),(61,99),(62,98),(63,97),(64,96),(65,95),(66,94),(67,93),(68,92),(69,91),(70,105),(71,104),(72,103),(73,102),(74,101),(75,100),(76,114),(77,113),(78,112),(79,111),(80,110),(81,109),(82,108),(83,107),(84,106),(85,120),(86,119),(87,118),(88,117),(89,116),(90,115),(121,147),(122,146),(123,145),(124,144),(125,143),(126,142),(127,141),(128,140),(129,139),(130,138),(131,137),(132,136),(133,150),(134,149),(135,148),(151,166),(152,180),(153,179),(154,178),(155,177),(156,176),(157,175),(158,174),(159,173),(160,172),(161,171),(162,170),(163,169),(164,168),(165,167),(181,237),(182,236),(183,235),(184,234),(185,233),(186,232),(187,231),(188,230),(189,229),(190,228),(191,227),(192,226),(193,240),(194,239),(195,238),(196,215),(197,214),(198,213),(199,212),(200,211),(201,225),(202,224),(203,223),(204,222),(205,221),(206,220),(207,219),(208,218),(209,217),(210,216)])`

84 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 6A 6B 6C 10A ··· 10F 12A ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 60 2 2 2 4 4 30 30 30 30 60 2 2 2 2 2 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + - + - + - + image C1 C2 C2 C2 C2 C2 S3 D5 D6 C4○D4 D10 D15 C4○D12 D30 C4○D20 D60⋊11C2 D4⋊2S3 Q8⋊3S3 D4⋊2D5 Q8⋊2D5 D4⋊2D15 Q8⋊3D15 kernel C4⋊C4⋊D15 C4×Dic15 C30.4Q8 C60⋊5C4 D30⋊3C4 C15×C4⋊C4 C5×C4⋊C4 C3×C4⋊C4 C2×C20 C30 C2×C12 C4⋊C4 C10 C2×C4 C6 C2 C10 C10 C6 C6 C2 C2 # reps 1 1 1 1 3 1 1 2 3 6 6 4 4 12 8 16 1 1 2 2 4 4

Matrix representation of C4⋊C4⋊D15 in GL8(𝔽61)

 44 56 0 0 0 0 0 0 21 17 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 50 0 0 0 0 0 0 0 29 11
,
 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 26 5 0 0 0 0 0 0 48 35
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 17 0 0 0 0 0 0 44 44 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 42 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 0 0 60 17 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 14 60

`G:=sub<GL(8,GF(61))| [44,21,0,0,0,0,0,0,56,17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,50,29,0,0,0,0,0,0,0,11],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,26,48,0,0,0,0,0,0,5,35],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,44,0,0,0,0,0,0,17,44,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,42,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,17,1,0,0,0,0,0,0,0,0,1,14,0,0,0,0,0,0,0,60] >;`

C4⋊C4⋊D15 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4\rtimes D_{15}`
`% in TeX`

`G:=Group("C4:C4:D15");`
`// GroupNames label`

`G:=SmallGroup(480,863);`
`// by ID`

`G=gap.SmallGroup(480,863);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,254,219,100,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^15=d^2=1,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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