direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4×D15, D30.3Q8, D30.47D4, C20⋊9(C4×S3), C4⋊3(C4×D15), C12⋊5(C4×D5), C60⋊11(C2×C4), (C4×D15)⋊5C4, C2.3(D4×D15), C2.2(Q8×D15), C6.41(Q8×D5), C60⋊5C4⋊17C2, (C2×C4).29D30, C6.103(D4×D5), C30.94(C2×Q8), C10.41(S3×Q8), D30.43(C2×C4), C10.105(S3×D4), (C2×C20).209D6, C30.311(C2×D4), Dic15⋊19(C2×C4), (C2×C12).207D10, (C2×C60).64C22, C30.4Q8⋊11C2, C30.160(C22×C4), (C2×C30).288C23, C22.16(C22×D15), (C2×Dic15).162C22, (C22×D15).126C22, C5⋊4(S3×C4⋊C4), C3⋊3(D5×C4⋊C4), C15⋊14(C2×C4⋊C4), (C5×C4⋊C4)⋊2S3, (C3×C4⋊C4)⋊2D5, (C15×C4⋊C4)⋊2C2, C6.65(C2×C4×D5), C10.97(S3×C2×C4), C2.11(C2×C4×D15), (C2×C4×D15).10C2, (C2×C6).284(C22×D5), (C2×C10).283(C22×S3), SmallGroup(480,856)
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, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 996 in 184 conjugacy classes, 71 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, C4⋊C4, C4⋊C4, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C2×C4⋊C4, C4×D5, C2×Dic5, C2×C20, C2×C20, C22×D5, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, Dic15, Dic15, C60, C60, D30, C2×C30, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C4×D5, S3×C4⋊C4, C4×D15, C4×D15, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×D15, D5×C4⋊C4, C30.4Q8, C60⋊5C4, C15×C4⋊C4, C2×C4×D15, C2×C4×D15, C4⋊C4×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D5, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C4×S3, C22×S3, D15, C2×C4⋊C4, C4×D5, C22×D5, S3×C2×C4, S3×D4, S3×Q8, D30, C2×C4×D5, D4×D5, Q8×D5, S3×C4⋊C4, C4×D15, C22×D15, D5×C4⋊C4, C2×C4×D15, D4×D15, Q8×D15, C4⋊C4×D15
►On 240 points
Generators in S240
(1 170 24 159)(2 171 25 160)(3 172 26 161)(4 173 27 162)(5 174 28 163)(6 175 29 164)(7 176 30 165)(8 177 16 151)(9 178 17 152)(10 179 18 153)(11 180 19 154)(12 166 20 155)(13 167 21 156)(14 168 22 157)(15 169 23 158)(31 144 57 121)(32 145 58 122)(33 146 59 123)(34 147 60 124)(35 148 46 125)(36 149 47 126)(37 150 48 127)(38 136 49 128)(39 137 50 129)(40 138 51 130)(41 139 52 131)(42 140 53 132)(43 141 54 133)(44 142 55 134)(45 143 56 135)(61 237 83 215)(62 238 84 216)(63 239 85 217)(64 240 86 218)(65 226 87 219)(66 227 88 220)(67 228 89 221)(68 229 90 222)(69 230 76 223)(70 231 77 224)(71 232 78 225)(72 233 79 211)(73 234 80 212)(74 235 81 213)(75 236 82 214)(91 203 106 184)(92 204 107 185)(93 205 108 186)(94 206 109 187)(95 207 110 188)(96 208 111 189)(97 209 112 190)(98 210 113 191)(99 196 114 192)(100 197 115 193)(101 198 116 194)(102 199 117 195)(103 200 118 181)(104 201 119 182)(105 202 120 183)
(1 92 35 69)(2 93 36 70)(3 94 37 71)(4 95 38 72)(5 96 39 73)(6 97 40 74)(7 98 41 75)(8 99 42 61)(9 100 43 62)(10 101 44 63)(11 102 45 64)(12 103 31 65)(13 104 32 66)(14 105 33 67)(15 91 34 68)(16 114 53 83)(17 115 54 84)(18 116 55 85)(19 117 56 86)(20 118 57 87)(21 119 58 88)(22 120 59 89)(23 106 60 90)(24 107 46 76)(25 108 47 77)(26 109 48 78)(27 110 49 79)(28 111 50 80)(29 112 51 81)(30 113 52 82)(121 226 155 200)(122 227 156 201)(123 228 157 202)(124 229 158 203)(125 230 159 204)(126 231 160 205)(127 232 161 206)(128 233 162 207)(129 234 163 208)(130 235 164 209)(131 236 165 210)(132 237 151 196)(133 238 152 197)(134 239 153 198)(135 240 154 199)(136 211 173 188)(137 212 174 189)(138 213 175 190)(139 214 176 191)(140 215 177 192)(141 216 178 193)(142 217 179 194)(143 218 180 195)(144 219 166 181)(145 220 167 182)(146 221 168 183)(147 222 169 184)(148 223 170 185)(149 224 171 186)(150 225 172 187)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(31 49)(32 48)(33 47)(34 46)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(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)(91 107)(92 106)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)(121 136)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)(134 138)(135 137)(151 177)(152 176)(153 175)(154 174)(155 173)(156 172)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 180)(164 179)(165 178)(181 207)(182 206)(183 205)(184 204)(185 203)(186 202)(187 201)(188 200)(189 199)(190 198)(191 197)(192 196)(193 210)(194 209)(195 208)(211 226)(212 240)(213 239)(214 238)(215 237)(216 236)(217 235)(218 234)(219 233)(220 232)(221 231)(222 230)(223 229)(224 228)(225 227)
G:=sub<Sym(240)| (1,170,24,159)(2,171,25,160)(3,172,26,161)(4,173,27,162)(5,174,28,163)(6,175,29,164)(7,176,30,165)(8,177,16,151)(9,178,17,152)(10,179,18,153)(11,180,19,154)(12,166,20,155)(13,167,21,156)(14,168,22,157)(15,169,23,158)(31,144,57,121)(32,145,58,122)(33,146,59,123)(34,147,60,124)(35,148,46,125)(36,149,47,126)(37,150,48,127)(38,136,49,128)(39,137,50,129)(40,138,51,130)(41,139,52,131)(42,140,53,132)(43,141,54,133)(44,142,55,134)(45,143,56,135)(61,237,83,215)(62,238,84,216)(63,239,85,217)(64,240,86,218)(65,226,87,219)(66,227,88,220)(67,228,89,221)(68,229,90,222)(69,230,76,223)(70,231,77,224)(71,232,78,225)(72,233,79,211)(73,234,80,212)(74,235,81,213)(75,236,82,214)(91,203,106,184)(92,204,107,185)(93,205,108,186)(94,206,109,187)(95,207,110,188)(96,208,111,189)(97,209,112,190)(98,210,113,191)(99,196,114,192)(100,197,115,193)(101,198,116,194)(102,199,117,195)(103,200,118,181)(104,201,119,182)(105,202,120,183), (1,92,35,69)(2,93,36,70)(3,94,37,71)(4,95,38,72)(5,96,39,73)(6,97,40,74)(7,98,41,75)(8,99,42,61)(9,100,43,62)(10,101,44,63)(11,102,45,64)(12,103,31,65)(13,104,32,66)(14,105,33,67)(15,91,34,68)(16,114,53,83)(17,115,54,84)(18,116,55,85)(19,117,56,86)(20,118,57,87)(21,119,58,88)(22,120,59,89)(23,106,60,90)(24,107,46,76)(25,108,47,77)(26,109,48,78)(27,110,49,79)(28,111,50,80)(29,112,51,81)(30,113,52,82)(121,226,155,200)(122,227,156,201)(123,228,157,202)(124,229,158,203)(125,230,159,204)(126,231,160,205)(127,232,161,206)(128,233,162,207)(129,234,163,208)(130,235,164,209)(131,236,165,210)(132,237,151,196)(133,238,152,197)(134,239,153,198)(135,240,154,199)(136,211,173,188)(137,212,174,189)(138,213,175,190)(139,214,176,191)(140,215,177,192)(141,216,178,193)(142,217,179,194)(143,218,180,195)(144,219,166,181)(145,220,167,182)(146,221,168,183)(147,222,169,184)(148,223,170,185)(149,224,171,186)(150,225,172,187), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(31,49)(32,48)(33,47)(34,46)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(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)(91,107)(92,106)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(121,136)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,180)(164,179)(165,178)(181,207)(182,206)(183,205)(184,204)(185,203)(186,202)(187,201)(188,200)(189,199)(190,198)(191,197)(192,196)(193,210)(194,209)(195,208)(211,226)(212,240)(213,239)(214,238)(215,237)(216,236)(217,235)(218,234)(219,233)(220,232)(221,231)(222,230)(223,229)(224,228)(225,227)>;
G:=Group( (1,170,24,159)(2,171,25,160)(3,172,26,161)(4,173,27,162)(5,174,28,163)(6,175,29,164)(7,176,30,165)(8,177,16,151)(9,178,17,152)(10,179,18,153)(11,180,19,154)(12,166,20,155)(13,167,21,156)(14,168,22,157)(15,169,23,158)(31,144,57,121)(32,145,58,122)(33,146,59,123)(34,147,60,124)(35,148,46,125)(36,149,47,126)(37,150,48,127)(38,136,49,128)(39,137,50,129)(40,138,51,130)(41,139,52,131)(42,140,53,132)(43,141,54,133)(44,142,55,134)(45,143,56,135)(61,237,83,215)(62,238,84,216)(63,239,85,217)(64,240,86,218)(65,226,87,219)(66,227,88,220)(67,228,89,221)(68,229,90,222)(69,230,76,223)(70,231,77,224)(71,232,78,225)(72,233,79,211)(73,234,80,212)(74,235,81,213)(75,236,82,214)(91,203,106,184)(92,204,107,185)(93,205,108,186)(94,206,109,187)(95,207,110,188)(96,208,111,189)(97,209,112,190)(98,210,113,191)(99,196,114,192)(100,197,115,193)(101,198,116,194)(102,199,117,195)(103,200,118,181)(104,201,119,182)(105,202,120,183), (1,92,35,69)(2,93,36,70)(3,94,37,71)(4,95,38,72)(5,96,39,73)(6,97,40,74)(7,98,41,75)(8,99,42,61)(9,100,43,62)(10,101,44,63)(11,102,45,64)(12,103,31,65)(13,104,32,66)(14,105,33,67)(15,91,34,68)(16,114,53,83)(17,115,54,84)(18,116,55,85)(19,117,56,86)(20,118,57,87)(21,119,58,88)(22,120,59,89)(23,106,60,90)(24,107,46,76)(25,108,47,77)(26,109,48,78)(27,110,49,79)(28,111,50,80)(29,112,51,81)(30,113,52,82)(121,226,155,200)(122,227,156,201)(123,228,157,202)(124,229,158,203)(125,230,159,204)(126,231,160,205)(127,232,161,206)(128,233,162,207)(129,234,163,208)(130,235,164,209)(131,236,165,210)(132,237,151,196)(133,238,152,197)(134,239,153,198)(135,240,154,199)(136,211,173,188)(137,212,174,189)(138,213,175,190)(139,214,176,191)(140,215,177,192)(141,216,178,193)(142,217,179,194)(143,218,180,195)(144,219,166,181)(145,220,167,182)(146,221,168,183)(147,222,169,184)(148,223,170,185)(149,224,171,186)(150,225,172,187), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(31,49)(32,48)(33,47)(34,46)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(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)(91,107)(92,106)(93,120)(94,119)(95,118)(96,117)(97,116)(98,115)(99,114)(100,113)(101,112)(102,111)(103,110)(104,109)(105,108)(121,136)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)(134,138)(135,137)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,180)(164,179)(165,178)(181,207)(182,206)(183,205)(184,204)(185,203)(186,202)(187,201)(188,200)(189,199)(190,198)(191,197)(192,196)(193,210)(194,209)(195,208)(211,226)(212,240)(213,239)(214,238)(215,237)(216,236)(217,235)(218,234)(219,233)(220,232)(221,231)(222,230)(223,229)(224,228)(225,227) );
G=PermutationGroup([[(1,170,24,159),(2,171,25,160),(3,172,26,161),(4,173,27,162),(5,174,28,163),(6,175,29,164),(7,176,30,165),(8,177,16,151),(9,178,17,152),(10,179,18,153),(11,180,19,154),(12,166,20,155),(13,167,21,156),(14,168,22,157),(15,169,23,158),(31,144,57,121),(32,145,58,122),(33,146,59,123),(34,147,60,124),(35,148,46,125),(36,149,47,126),(37,150,48,127),(38,136,49,128),(39,137,50,129),(40,138,51,130),(41,139,52,131),(42,140,53,132),(43,141,54,133),(44,142,55,134),(45,143,56,135),(61,237,83,215),(62,238,84,216),(63,239,85,217),(64,240,86,218),(65,226,87,219),(66,227,88,220),(67,228,89,221),(68,229,90,222),(69,230,76,223),(70,231,77,224),(71,232,78,225),(72,233,79,211),(73,234,80,212),(74,235,81,213),(75,236,82,214),(91,203,106,184),(92,204,107,185),(93,205,108,186),(94,206,109,187),(95,207,110,188),(96,208,111,189),(97,209,112,190),(98,210,113,191),(99,196,114,192),(100,197,115,193),(101,198,116,194),(102,199,117,195),(103,200,118,181),(104,201,119,182),(105,202,120,183)], [(1,92,35,69),(2,93,36,70),(3,94,37,71),(4,95,38,72),(5,96,39,73),(6,97,40,74),(7,98,41,75),(8,99,42,61),(9,100,43,62),(10,101,44,63),(11,102,45,64),(12,103,31,65),(13,104,32,66),(14,105,33,67),(15,91,34,68),(16,114,53,83),(17,115,54,84),(18,116,55,85),(19,117,56,86),(20,118,57,87),(21,119,58,88),(22,120,59,89),(23,106,60,90),(24,107,46,76),(25,108,47,77),(26,109,48,78),(27,110,49,79),(28,111,50,80),(29,112,51,81),(30,113,52,82),(121,226,155,200),(122,227,156,201),(123,228,157,202),(124,229,158,203),(125,230,159,204),(126,231,160,205),(127,232,161,206),(128,233,162,207),(129,234,163,208),(130,235,164,209),(131,236,165,210),(132,237,151,196),(133,238,152,197),(134,239,153,198),(135,240,154,199),(136,211,173,188),(137,212,174,189),(138,213,175,190),(139,214,176,191),(140,215,177,192),(141,216,178,193),(142,217,179,194),(143,218,180,195),(144,219,166,181),(145,220,167,182),(146,221,168,183),(147,222,169,184),(148,223,170,185),(149,224,171,186),(150,225,172,187)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(31,49),(32,48),(33,47),(34,46),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(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),(91,107),(92,106),(93,120),(94,119),(95,118),(96,117),(97,116),(98,115),(99,114),(100,113),(101,112),(102,111),(103,110),(104,109),(105,108),(121,136),(122,150),(123,149),(124,148),(125,147),(126,146),(127,145),(128,144),(129,143),(130,142),(131,141),(132,140),(133,139),(134,138),(135,137),(151,177),(152,176),(153,175),(154,174),(155,173),(156,172),(157,171),(158,170),(159,169),(160,168),(161,167),(162,166),(163,180),(164,179),(165,178),(181,207),(182,206),(183,205),(184,204),(185,203),(186,202),(187,201),(188,200),(189,199),(190,198),(191,197),(192,196),(193,210),(194,209),(195,208),(211,226),(212,240),(213,239),(214,238),(215,237),(216,236),(217,235),(218,234),(219,233),(220,232),(221,231),(222,230),(223,229),(224,228),(225,227)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | ··· | 4L | 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 | 2 | 2 | 3 | 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 | 15 | 15 | 15 | 15 | 2 | 2 | ··· | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D5 | D6 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 | S3×D4 | S3×Q8 | D4×D5 | Q8×D5 | D4×D15 | Q8×D15 |
| kernel | C4⋊C4×D15 | C30.4Q8 | C60⋊5C4 | C15×C4⋊C4 | C2×C4×D15 | C4×D15 | C5×C4⋊C4 | D30 | D30 | C3×C4⋊C4 | C2×C20 | C2×C12 | C20 | C4⋊C4 | C12 | C2×C4 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 8 | 1 | 2 | 2 | 2 | 3 | 6 | 4 | 4 | 8 | 12 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C4⋊C4×D15 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 51 | 19 |
| 0 | 0 | 30 | 10 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 1 | 59 |
| 0 | 0 | 1 | 60 |
| 47 | 30 | 0 | 0 |
| 31 | 25 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 33 | 37 | 0 | 0 |
| 25 | 28 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,51,30,0,0,19,10],[11,0,0,0,0,11,0,0,0,0,1,1,0,0,59,60],[47,31,0,0,30,25,0,0,0,0,1,0,0,0,0,1],[33,25,0,0,37,28,0,0,0,0,60,0,0,0,0,60] >;
C4⋊C4×D15 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times D_{15} % in TeX
G:=Group("C4:C4xD15"); // GroupNames label
G:=SmallGroup(480,856);
// by ID
G=gap.SmallGroup(480,856);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,58,2693,18822]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations