metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊6D15, C60⋊5C4⋊13C2, (C2×C4).31D30, (C2×C20).212D6, (C4×Dic15)⋊20C2, D30⋊3C4.4C2, (C2×C12).248D10, C30.4Q8⋊35C2, C15⋊21(C42⋊2C2), C6.102(C4○D20), C30.221(C4○D4), C2.7(Q8⋊3D15), (C2×C30).295C23, (C2×C60).432C22, C6.43(Q8⋊2D5), C10.102(C4○D12), C6.100(D4⋊2D5), C2.14(D4⋊2D15), C10.43(Q8⋊3S3), C2.16(D60⋊11C2), C10.100(D4⋊2S3), (C22×D15).9C22, C22.53(C22×D15), (C2×Dic15).166C22, (C5×C4⋊C4)⋊9S3, (C3×C4⋊C4)⋊9D5, (C15×C4⋊C4)⋊9C2, C5⋊7(C4⋊C4⋊S3), C3⋊7(C4⋊C4⋊D5), (C2×C6).291(C22×D5), (C2×C10).290(C22×S3), SmallGroup(480,863)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C4⋊C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×C12, C22×S3, D15, C30, C42⋊2C2, C2×Dic5, C2×C20, C22×D5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, Dic15, C60, D30, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C4⋊C4⋊S3, C2×Dic15, C2×C60, C22×D15, C4⋊C4⋊D5, C4×Dic15, C30.4Q8, C60⋊5C4, D30⋊3C4, C15×C4⋊C4, C4⋊C4⋊D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, D15, C42⋊2C2, C22×D5, C4○D12, D4⋊2S3, Q8⋊3S3, D30, C4○D20, D4⋊2D5, Q8⋊2D5, C4⋊C4⋊S3, C22×D15, C4⋊C4⋊D5, D60⋊11C2, D4⋊2D15, Q8⋊3D15, C4⋊C4⋊D15
►On 240 points
Generators in S240
(1 114 20 103)(2 115 21 104)(3 116 22 105)(4 117 23 91)(5 118 24 92)(6 119 25 93)(7 120 26 94)(8 106 27 95)(9 107 28 96)(10 108 29 97)(11 109 30 98)(12 110 16 99)(13 111 17 100)(14 112 18 101)(15 113 19 102)(31 83 53 65)(32 84 54 66)(33 85 55 67)(34 86 56 68)(35 87 57 69)(36 88 58 70)(37 89 59 71)(38 90 60 72)(39 76 46 73)(40 77 47 74)(41 78 48 75)(42 79 49 61)(43 80 50 62)(44 81 51 63)(45 82 52 64)(121 211 143 237)(122 212 144 238)(123 213 145 239)(124 214 146 240)(125 215 147 226)(126 216 148 227)(127 217 149 228)(128 218 150 229)(129 219 136 230)(130 220 137 231)(131 221 138 232)(132 222 139 233)(133 223 140 234)(134 224 141 235)(135 225 142 236)(151 192 166 204)(152 193 167 205)(153 194 168 206)(154 195 169 207)(155 181 170 208)(156 182 171 209)(157 183 172 210)(158 184 173 196)(159 185 174 197)(160 186 175 198)(161 187 176 199)(162 188 177 200)(163 189 178 201)(164 190 179 202)(165 191 180 203)
(1 159 39 122)(2 160 40 123)(3 161 41 124)(4 162 42 125)(5 163 43 126)(6 164 44 127)(7 165 45 128)(8 151 31 129)(9 152 32 130)(10 153 33 131)(11 154 34 132)(12 155 35 133)(13 156 36 134)(14 157 37 135)(15 158 38 121)(16 170 57 140)(17 171 58 141)(18 172 59 142)(19 173 60 143)(20 174 46 144)(21 175 47 145)(22 176 48 146)(23 177 49 147)(24 178 50 148)(25 179 51 149)(26 180 52 150)(27 166 53 136)(28 167 54 137)(29 168 55 138)(30 169 56 139)(61 215 91 188)(62 216 92 189)(63 217 93 190)(64 218 94 191)(65 219 95 192)(66 220 96 193)(67 221 97 194)(68 222 98 195)(69 223 99 181)(70 224 100 182)(71 225 101 183)(72 211 102 184)(73 212 103 185)(74 213 104 186)(75 214 105 187)(76 238 114 197)(77 239 115 198)(78 240 116 199)(79 226 117 200)(80 227 118 201)(81 228 119 202)(82 229 120 203)(83 230 106 204)(84 231 107 205)(85 232 108 206)(86 233 109 207)(87 234 110 208)(88 235 111 209)(89 236 112 210)(90 237 113 196)
(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)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(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)(76 113)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(121 144)(122 143)(123 142)(124 141)(125 140)(126 139)(127 138)(128 137)(129 136)(130 150)(131 149)(132 148)(133 147)(134 146)(135 145)(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)(181 226)(182 240)(183 239)(184 238)(185 237)(186 236)(187 235)(188 234)(189 233)(190 232)(191 231)(192 230)(193 229)(194 228)(195 227)(196 212)(197 211)(198 225)(199 224)(200 223)(201 222)(202 221)(203 220)(204 219)(205 218)(206 217)(207 216)(208 215)(209 214)(210 213)
G:=sub<Sym(240)| (1,114,20,103)(2,115,21,104)(3,116,22,105)(4,117,23,91)(5,118,24,92)(6,119,25,93)(7,120,26,94)(8,106,27,95)(9,107,28,96)(10,108,29,97)(11,109,30,98)(12,110,16,99)(13,111,17,100)(14,112,18,101)(15,113,19,102)(31,83,53,65)(32,84,54,66)(33,85,55,67)(34,86,56,68)(35,87,57,69)(36,88,58,70)(37,89,59,71)(38,90,60,72)(39,76,46,73)(40,77,47,74)(41,78,48,75)(42,79,49,61)(43,80,50,62)(44,81,51,63)(45,82,52,64)(121,211,143,237)(122,212,144,238)(123,213,145,239)(124,214,146,240)(125,215,147,226)(126,216,148,227)(127,217,149,228)(128,218,150,229)(129,219,136,230)(130,220,137,231)(131,221,138,232)(132,222,139,233)(133,223,140,234)(134,224,141,235)(135,225,142,236)(151,192,166,204)(152,193,167,205)(153,194,168,206)(154,195,169,207)(155,181,170,208)(156,182,171,209)(157,183,172,210)(158,184,173,196)(159,185,174,197)(160,186,175,198)(161,187,176,199)(162,188,177,200)(163,189,178,201)(164,190,179,202)(165,191,180,203), (1,159,39,122)(2,160,40,123)(3,161,41,124)(4,162,42,125)(5,163,43,126)(6,164,44,127)(7,165,45,128)(8,151,31,129)(9,152,32,130)(10,153,33,131)(11,154,34,132)(12,155,35,133)(13,156,36,134)(14,157,37,135)(15,158,38,121)(16,170,57,140)(17,171,58,141)(18,172,59,142)(19,173,60,143)(20,174,46,144)(21,175,47,145)(22,176,48,146)(23,177,49,147)(24,178,50,148)(25,179,51,149)(26,180,52,150)(27,166,53,136)(28,167,54,137)(29,168,55,138)(30,169,56,139)(61,215,91,188)(62,216,92,189)(63,217,93,190)(64,218,94,191)(65,219,95,192)(66,220,96,193)(67,221,97,194)(68,222,98,195)(69,223,99,181)(70,224,100,182)(71,225,101,183)(72,211,102,184)(73,212,103,185)(74,213,104,186)(75,214,105,187)(76,238,114,197)(77,239,115,198)(78,240,116,199)(79,226,117,200)(80,227,118,201)(81,228,119,202)(82,229,120,203)(83,230,106,204)(84,231,107,205)(85,232,108,206)(86,233,109,207)(87,234,110,208)(88,235,111,209)(89,236,112,210)(90,237,113,196), (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)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(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)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,138)(128,137)(129,136)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(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)(181,226)(182,240)(183,239)(184,238)(185,237)(186,236)(187,235)(188,234)(189,233)(190,232)(191,231)(192,230)(193,229)(194,228)(195,227)(196,212)(197,211)(198,225)(199,224)(200,223)(201,222)(202,221)(203,220)(204,219)(205,218)(206,217)(207,216)(208,215)(209,214)(210,213)>;
G:=Group( (1,114,20,103)(2,115,21,104)(3,116,22,105)(4,117,23,91)(5,118,24,92)(6,119,25,93)(7,120,26,94)(8,106,27,95)(9,107,28,96)(10,108,29,97)(11,109,30,98)(12,110,16,99)(13,111,17,100)(14,112,18,101)(15,113,19,102)(31,83,53,65)(32,84,54,66)(33,85,55,67)(34,86,56,68)(35,87,57,69)(36,88,58,70)(37,89,59,71)(38,90,60,72)(39,76,46,73)(40,77,47,74)(41,78,48,75)(42,79,49,61)(43,80,50,62)(44,81,51,63)(45,82,52,64)(121,211,143,237)(122,212,144,238)(123,213,145,239)(124,214,146,240)(125,215,147,226)(126,216,148,227)(127,217,149,228)(128,218,150,229)(129,219,136,230)(130,220,137,231)(131,221,138,232)(132,222,139,233)(133,223,140,234)(134,224,141,235)(135,225,142,236)(151,192,166,204)(152,193,167,205)(153,194,168,206)(154,195,169,207)(155,181,170,208)(156,182,171,209)(157,183,172,210)(158,184,173,196)(159,185,174,197)(160,186,175,198)(161,187,176,199)(162,188,177,200)(163,189,178,201)(164,190,179,202)(165,191,180,203), (1,159,39,122)(2,160,40,123)(3,161,41,124)(4,162,42,125)(5,163,43,126)(6,164,44,127)(7,165,45,128)(8,151,31,129)(9,152,32,130)(10,153,33,131)(11,154,34,132)(12,155,35,133)(13,156,36,134)(14,157,37,135)(15,158,38,121)(16,170,57,140)(17,171,58,141)(18,172,59,142)(19,173,60,143)(20,174,46,144)(21,175,47,145)(22,176,48,146)(23,177,49,147)(24,178,50,148)(25,179,51,149)(26,180,52,150)(27,166,53,136)(28,167,54,137)(29,168,55,138)(30,169,56,139)(61,215,91,188)(62,216,92,189)(63,217,93,190)(64,218,94,191)(65,219,95,192)(66,220,96,193)(67,221,97,194)(68,222,98,195)(69,223,99,181)(70,224,100,182)(71,225,101,183)(72,211,102,184)(73,212,103,185)(74,213,104,186)(75,214,105,187)(76,238,114,197)(77,239,115,198)(78,240,116,199)(79,226,117,200)(80,227,118,201)(81,228,119,202)(82,229,120,203)(83,230,106,204)(84,231,107,205)(85,232,108,206)(86,233,109,207)(87,234,110,208)(88,235,111,209)(89,236,112,210)(90,237,113,196), (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)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(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)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,138)(128,137)(129,136)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)(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)(181,226)(182,240)(183,239)(184,238)(185,237)(186,236)(187,235)(188,234)(189,233)(190,232)(191,231)(192,230)(193,229)(194,228)(195,227)(196,212)(197,211)(198,225)(199,224)(200,223)(201,222)(202,221)(203,220)(204,219)(205,218)(206,217)(207,216)(208,215)(209,214)(210,213) );
G=PermutationGroup([[(1,114,20,103),(2,115,21,104),(3,116,22,105),(4,117,23,91),(5,118,24,92),(6,119,25,93),(7,120,26,94),(8,106,27,95),(9,107,28,96),(10,108,29,97),(11,109,30,98),(12,110,16,99),(13,111,17,100),(14,112,18,101),(15,113,19,102),(31,83,53,65),(32,84,54,66),(33,85,55,67),(34,86,56,68),(35,87,57,69),(36,88,58,70),(37,89,59,71),(38,90,60,72),(39,76,46,73),(40,77,47,74),(41,78,48,75),(42,79,49,61),(43,80,50,62),(44,81,51,63),(45,82,52,64),(121,211,143,237),(122,212,144,238),(123,213,145,239),(124,214,146,240),(125,215,147,226),(126,216,148,227),(127,217,149,228),(128,218,150,229),(129,219,136,230),(130,220,137,231),(131,221,138,232),(132,222,139,233),(133,223,140,234),(134,224,141,235),(135,225,142,236),(151,192,166,204),(152,193,167,205),(153,194,168,206),(154,195,169,207),(155,181,170,208),(156,182,171,209),(157,183,172,210),(158,184,173,196),(159,185,174,197),(160,186,175,198),(161,187,176,199),(162,188,177,200),(163,189,178,201),(164,190,179,202),(165,191,180,203)], [(1,159,39,122),(2,160,40,123),(3,161,41,124),(4,162,42,125),(5,163,43,126),(6,164,44,127),(7,165,45,128),(8,151,31,129),(9,152,32,130),(10,153,33,131),(11,154,34,132),(12,155,35,133),(13,156,36,134),(14,157,37,135),(15,158,38,121),(16,170,57,140),(17,171,58,141),(18,172,59,142),(19,173,60,143),(20,174,46,144),(21,175,47,145),(22,176,48,146),(23,177,49,147),(24,178,50,148),(25,179,51,149),(26,180,52,150),(27,166,53,136),(28,167,54,137),(29,168,55,138),(30,169,56,139),(61,215,91,188),(62,216,92,189),(63,217,93,190),(64,218,94,191),(65,219,95,192),(66,220,96,193),(67,221,97,194),(68,222,98,195),(69,223,99,181),(70,224,100,182),(71,225,101,183),(72,211,102,184),(73,212,103,185),(74,213,104,186),(75,214,105,187),(76,238,114,197),(77,239,115,198),(78,240,116,199),(79,226,117,200),(80,227,118,201),(81,228,119,202),(82,229,120,203),(83,230,106,204),(84,231,107,205),(85,232,108,206),(86,233,109,207),(87,234,110,208),(88,235,111,209),(89,236,112,210),(90,237,113,196)], [(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),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(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),(76,113),(77,112),(78,111),(79,110),(80,109),(81,108),(82,107),(83,106),(84,120),(85,119),(86,118),(87,117),(88,116),(89,115),(90,114),(121,144),(122,143),(123,142),(124,141),(125,140),(126,139),(127,138),(128,137),(129,136),(130,150),(131,149),(132,148),(133,147),(134,146),(135,145),(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),(181,226),(182,240),(183,239),(184,238),(185,237),(186,236),(187,235),(188,234),(189,233),(190,232),(191,231),(192,230),(193,229),(194,228),(195,227),(196,212),(197,211),(198,225),(199,224),(200,223),(201,222),(202,221),(203,220),(204,219),(205,218),(206,217),(207,216),(208,215),(209,214),(210,213)]])
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