metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊7D15, (C4×D15)⋊6C4, C20.60(C4×S3), C60.85(C2×C4), C60⋊5C4⋊18C2, (C2×C4).42D30, C12.28(C4×D5), C4.14(C4×D15), D30.34(C2×C4), (C2×C20).210D6, (C4×Dic15)⋊19C2, D30⋊3C4.5C2, (C2×C12).208D10, (C2×C60).65C22, C15⋊27(C42⋊C2), C30.257(C4○D4), C2.4(D4⋊2D15), C2.1(Q8⋊3D15), C6.98(D4⋊2D5), (C2×C30).289C23, C30.161(C22×C4), C6.39(Q8⋊2D5), Dic15.50(C2×C4), C10.98(D4⋊2S3), C10.39(Q8⋊3S3), C22.17(C22×D15), (C22×D15).81C22, (C2×Dic15).163C22, (C5×C4⋊C4)⋊3S3, (C3×C4⋊C4)⋊3D5, (C15×C4⋊C4)⋊3C2, C6.66(C2×C4×D5), C10.98(S3×C2×C4), (C2×C4×D15).1C2, C5⋊5(C4⋊C4⋊7S3), C2.12(C2×C4×D15), C3⋊4(C4⋊C4⋊7D5), (C2×C6).285(C22×D5), (C2×C10).284(C22×S3), SmallGroup(480,857)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊7D15
G = < a,b,c,d | a4=b4=c15=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 804 in 152 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C42⋊C2, C4×D5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, Dic15, Dic15, C60, C60, D30, D30, C2×C30, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C4⋊C4⋊7S3, C4×D15, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×D15, C4⋊C4⋊7D5, C4×Dic15, C60⋊5C4, D30⋊3C4, C15×C4⋊C4, C2×C4×D15, C4⋊C4⋊7D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, D10, C4×S3, C22×S3, D15, C42⋊C2, C4×D5, C22×D5, S3×C2×C4, D4⋊2S3, Q8⋊3S3, D30, C2×C4×D5, D4⋊2D5, Q8⋊2D5, C4⋊C4⋊7S3, C4×D15, C22×D15, C4⋊C4⋊7D5, C2×C4×D15, D4⋊2D15, Q8⋊3D15, C4⋊C4⋊7D15
►On 240 points
Generators in S240
(1 174 20 155)(2 175 21 156)(3 176 22 157)(4 177 23 158)(5 178 24 159)(6 179 25 160)(7 180 26 161)(8 166 27 162)(9 167 28 163)(10 168 29 164)(11 169 30 165)(12 170 16 151)(13 171 17 152)(14 172 18 153)(15 173 19 154)(31 143 46 125)(32 144 47 126)(33 145 48 127)(34 146 49 128)(35 147 50 129)(36 148 51 130)(37 149 52 131)(38 150 53 132)(39 136 54 133)(40 137 55 134)(41 138 56 135)(42 139 57 121)(43 140 58 122)(44 141 59 123)(45 142 60 124)(61 237 76 222)(62 238 77 223)(63 239 78 224)(64 240 79 225)(65 226 80 211)(66 227 81 212)(67 228 82 213)(68 229 83 214)(69 230 84 215)(70 231 85 216)(71 232 86 217)(72 233 87 218)(73 234 88 219)(74 235 89 220)(75 236 90 221)(91 207 106 185)(92 208 107 186)(93 209 108 187)(94 210 109 188)(95 196 110 189)(96 197 111 190)(97 198 112 191)(98 199 113 192)(99 200 114 193)(100 201 115 194)(101 202 116 195)(102 203 117 181)(103 204 118 182)(104 205 119 183)(105 206 120 184)
(1 99 43 65)(2 100 44 66)(3 101 45 67)(4 102 31 68)(5 103 32 69)(6 104 33 70)(7 105 34 71)(8 91 35 72)(9 92 36 73)(10 93 37 74)(11 94 38 75)(12 95 39 61)(13 96 40 62)(14 97 41 63)(15 98 42 64)(16 110 54 76)(17 111 55 77)(18 112 56 78)(19 113 57 79)(20 114 58 80)(21 115 59 81)(22 116 60 82)(23 117 46 83)(24 118 47 84)(25 119 48 85)(26 120 49 86)(27 106 50 87)(28 107 51 88)(29 108 52 89)(30 109 53 90)(121 240 154 199)(122 226 155 200)(123 227 156 201)(124 228 157 202)(125 229 158 203)(126 230 159 204)(127 231 160 205)(128 232 161 206)(129 233 162 207)(130 234 163 208)(131 235 164 209)(132 236 165 210)(133 237 151 196)(134 238 152 197)(135 239 153 198)(136 222 170 189)(137 223 171 190)(138 224 172 191)(139 225 173 192)(140 211 174 193)(141 212 175 194)(142 213 176 195)(143 214 177 181)(144 215 178 182)(145 216 179 183)(146 217 180 184)(147 218 166 185)(148 219 167 186)(149 220 168 187)(150 221 169 188)
(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 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(91 106)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(121 122)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(136 143)(137 142)(138 141)(139 140)(144 150)(145 149)(146 148)(151 158)(152 157)(153 156)(154 155)(159 165)(160 164)(161 163)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)(181 196)(182 210)(183 209)(184 208)(185 207)(186 206)(187 205)(188 204)(189 203)(190 202)(191 201)(192 200)(193 199)(194 198)(195 197)(211 240)(212 239)(213 238)(214 237)(215 236)(216 235)(217 234)(218 233)(219 232)(220 231)(221 230)(222 229)(223 228)(224 227)(225 226)
G:=sub<Sym(240)| (1,174,20,155)(2,175,21,156)(3,176,22,157)(4,177,23,158)(5,178,24,159)(6,179,25,160)(7,180,26,161)(8,166,27,162)(9,167,28,163)(10,168,29,164)(11,169,30,165)(12,170,16,151)(13,171,17,152)(14,172,18,153)(15,173,19,154)(31,143,46,125)(32,144,47,126)(33,145,48,127)(34,146,49,128)(35,147,50,129)(36,148,51,130)(37,149,52,131)(38,150,53,132)(39,136,54,133)(40,137,55,134)(41,138,56,135)(42,139,57,121)(43,140,58,122)(44,141,59,123)(45,142,60,124)(61,237,76,222)(62,238,77,223)(63,239,78,224)(64,240,79,225)(65,226,80,211)(66,227,81,212)(67,228,82,213)(68,229,83,214)(69,230,84,215)(70,231,85,216)(71,232,86,217)(72,233,87,218)(73,234,88,219)(74,235,89,220)(75,236,90,221)(91,207,106,185)(92,208,107,186)(93,209,108,187)(94,210,109,188)(95,196,110,189)(96,197,111,190)(97,198,112,191)(98,199,113,192)(99,200,114,193)(100,201,115,194)(101,202,116,195)(102,203,117,181)(103,204,118,182)(104,205,119,183)(105,206,120,184), (1,99,43,65)(2,100,44,66)(3,101,45,67)(4,102,31,68)(5,103,32,69)(6,104,33,70)(7,105,34,71)(8,91,35,72)(9,92,36,73)(10,93,37,74)(11,94,38,75)(12,95,39,61)(13,96,40,62)(14,97,41,63)(15,98,42,64)(16,110,54,76)(17,111,55,77)(18,112,56,78)(19,113,57,79)(20,114,58,80)(21,115,59,81)(22,116,60,82)(23,117,46,83)(24,118,47,84)(25,119,48,85)(26,120,49,86)(27,106,50,87)(28,107,51,88)(29,108,52,89)(30,109,53,90)(121,240,154,199)(122,226,155,200)(123,227,156,201)(124,228,157,202)(125,229,158,203)(126,230,159,204)(127,231,160,205)(128,232,161,206)(129,233,162,207)(130,234,163,208)(131,235,164,209)(132,236,165,210)(133,237,151,196)(134,238,152,197)(135,239,153,198)(136,222,170,189)(137,223,171,190)(138,224,172,191)(139,225,173,192)(140,211,174,193)(141,212,175,194)(142,213,176,195)(143,214,177,181)(144,215,178,182)(145,216,179,183)(146,217,180,184)(147,218,166,185)(148,219,167,186)(149,220,168,187)(150,221,169,188), (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)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(136,143)(137,142)(138,141)(139,140)(144,150)(145,149)(146,148)(151,158)(152,157)(153,156)(154,155)(159,165)(160,164)(161,163)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,196)(182,210)(183,209)(184,208)(185,207)(186,206)(187,205)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226)>;
G:=Group( (1,174,20,155)(2,175,21,156)(3,176,22,157)(4,177,23,158)(5,178,24,159)(6,179,25,160)(7,180,26,161)(8,166,27,162)(9,167,28,163)(10,168,29,164)(11,169,30,165)(12,170,16,151)(13,171,17,152)(14,172,18,153)(15,173,19,154)(31,143,46,125)(32,144,47,126)(33,145,48,127)(34,146,49,128)(35,147,50,129)(36,148,51,130)(37,149,52,131)(38,150,53,132)(39,136,54,133)(40,137,55,134)(41,138,56,135)(42,139,57,121)(43,140,58,122)(44,141,59,123)(45,142,60,124)(61,237,76,222)(62,238,77,223)(63,239,78,224)(64,240,79,225)(65,226,80,211)(66,227,81,212)(67,228,82,213)(68,229,83,214)(69,230,84,215)(70,231,85,216)(71,232,86,217)(72,233,87,218)(73,234,88,219)(74,235,89,220)(75,236,90,221)(91,207,106,185)(92,208,107,186)(93,209,108,187)(94,210,109,188)(95,196,110,189)(96,197,111,190)(97,198,112,191)(98,199,113,192)(99,200,114,193)(100,201,115,194)(101,202,116,195)(102,203,117,181)(103,204,118,182)(104,205,119,183)(105,206,120,184), (1,99,43,65)(2,100,44,66)(3,101,45,67)(4,102,31,68)(5,103,32,69)(6,104,33,70)(7,105,34,71)(8,91,35,72)(9,92,36,73)(10,93,37,74)(11,94,38,75)(12,95,39,61)(13,96,40,62)(14,97,41,63)(15,98,42,64)(16,110,54,76)(17,111,55,77)(18,112,56,78)(19,113,57,79)(20,114,58,80)(21,115,59,81)(22,116,60,82)(23,117,46,83)(24,118,47,84)(25,119,48,85)(26,120,49,86)(27,106,50,87)(28,107,51,88)(29,108,52,89)(30,109,53,90)(121,240,154,199)(122,226,155,200)(123,227,156,201)(124,228,157,202)(125,229,158,203)(126,230,159,204)(127,231,160,205)(128,232,161,206)(129,233,162,207)(130,234,163,208)(131,235,164,209)(132,236,165,210)(133,237,151,196)(134,238,152,197)(135,239,153,198)(136,222,170,189)(137,223,171,190)(138,224,172,191)(139,225,173,192)(140,211,174,193)(141,212,175,194)(142,213,176,195)(143,214,177,181)(144,215,178,182)(145,216,179,183)(146,217,180,184)(147,218,166,185)(148,219,167,186)(149,220,168,187)(150,221,169,188), (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)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(136,143)(137,142)(138,141)(139,140)(144,150)(145,149)(146,148)(151,158)(152,157)(153,156)(154,155)(159,165)(160,164)(161,163)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,196)(182,210)(183,209)(184,208)(185,207)(186,206)(187,205)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226) );
G=PermutationGroup([[(1,174,20,155),(2,175,21,156),(3,176,22,157),(4,177,23,158),(5,178,24,159),(6,179,25,160),(7,180,26,161),(8,166,27,162),(9,167,28,163),(10,168,29,164),(11,169,30,165),(12,170,16,151),(13,171,17,152),(14,172,18,153),(15,173,19,154),(31,143,46,125),(32,144,47,126),(33,145,48,127),(34,146,49,128),(35,147,50,129),(36,148,51,130),(37,149,52,131),(38,150,53,132),(39,136,54,133),(40,137,55,134),(41,138,56,135),(42,139,57,121),(43,140,58,122),(44,141,59,123),(45,142,60,124),(61,237,76,222),(62,238,77,223),(63,239,78,224),(64,240,79,225),(65,226,80,211),(66,227,81,212),(67,228,82,213),(68,229,83,214),(69,230,84,215),(70,231,85,216),(71,232,86,217),(72,233,87,218),(73,234,88,219),(74,235,89,220),(75,236,90,221),(91,207,106,185),(92,208,107,186),(93,209,108,187),(94,210,109,188),(95,196,110,189),(96,197,111,190),(97,198,112,191),(98,199,113,192),(99,200,114,193),(100,201,115,194),(101,202,116,195),(102,203,117,181),(103,204,118,182),(104,205,119,183),(105,206,120,184)], [(1,99,43,65),(2,100,44,66),(3,101,45,67),(4,102,31,68),(5,103,32,69),(6,104,33,70),(7,105,34,71),(8,91,35,72),(9,92,36,73),(10,93,37,74),(11,94,38,75),(12,95,39,61),(13,96,40,62),(14,97,41,63),(15,98,42,64),(16,110,54,76),(17,111,55,77),(18,112,56,78),(19,113,57,79),(20,114,58,80),(21,115,59,81),(22,116,60,82),(23,117,46,83),(24,118,47,84),(25,119,48,85),(26,120,49,86),(27,106,50,87),(28,107,51,88),(29,108,52,89),(30,109,53,90),(121,240,154,199),(122,226,155,200),(123,227,156,201),(124,228,157,202),(125,229,158,203),(126,230,159,204),(127,231,160,205),(128,232,161,206),(129,233,162,207),(130,234,163,208),(131,235,164,209),(132,236,165,210),(133,237,151,196),(134,238,152,197),(135,239,153,198),(136,222,170,189),(137,223,171,190),(138,224,172,191),(139,225,173,192),(140,211,174,193),(141,212,175,194),(142,213,176,195),(143,214,177,181),(144,215,178,182),(145,216,179,183),(146,217,180,184),(147,218,166,185),(148,219,167,186),(149,220,168,187),(150,221,169,188)], [(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),(31,39),(32,38),(33,37),(34,36),(40,45),(41,44),(42,43),(46,54),(47,53),(48,52),(49,51),(55,60),(56,59),(57,58),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(91,106),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(121,122),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(136,143),(137,142),(138,141),(139,140),(144,150),(145,149),(146,148),(151,158),(152,157),(153,156),(154,155),(159,165),(160,164),(161,163),(167,180),(168,179),(169,178),(170,177),(171,176),(172,175),(173,174),(181,196),(182,210),(183,209),(184,208),(185,207),(186,206),(187,205),(188,204),(189,203),(190,202),(191,201),(192,200),(193,199),(194,198),(195,197),(211,240),(212,239),(213,238),(214,237),(215,236),(216,235),(217,234),(218,233),(219,232),(220,231),(221,230),(222,229),(223,228),(224,227),(225,226)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 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 | 2 | 3 | 4 | ··· | 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 | 30 | 30 | 2 | 2 | ··· | 2 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 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 | C4 | S3 | D5 | D6 | C4○D4 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 | D4⋊2S3 | Q8⋊3S3 | D4⋊2D5 | Q8⋊2D5 | D4⋊2D15 | Q8⋊3D15 |
| kernel | C4⋊C4⋊7D15 | C4×Dic15 | C60⋊5C4 | D30⋊3C4 | C15×C4⋊C4 | C2×C4×D15 | C4×D15 | C5×C4⋊C4 | C3×C4⋊C4 | C2×C20 | C30 | C2×C12 | C20 | C4⋊C4 | C12 | C2×C4 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 1 | 2 | 3 | 4 | 6 | 4 | 4 | 8 | 12 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C4⋊C4⋊7D15 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 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 | 1 | 0 |
| 43 | 17 | 0 | 0 | 0 | 0 |
| 43 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 30 | 0 | 0 |
| 0 | 0 | 31 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 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 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,11],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[43,43,0,0,0,0,17,0,0,0,0,0,0,0,47,31,0,0,0,0,30,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;
C4⋊C4⋊7D15 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_7D_{15} % in TeX
G:=Group("C4:C4:7D15"); // GroupNames label
G:=SmallGroup(480,857);
// by ID
G=gap.SmallGroup(480,857);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,219,58,2693,18822]);
// 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,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations