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G = C4⋊C4×D15order 480 = 25·3·5

Direct product of C4⋊C4 and D15

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4×D15, D30.3Q8, D30.47D4, C209(C4×S3), C43(C4×D15), C125(C4×D5), C6011(C2×C4), (C4×D15)⋊5C4, C2.3(D4×D15), C2.2(Q8×D15), C6.41(Q8×D5), C605C417C2, (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), Dic1519(C2×C4), (C2×C12).207D10, (C2×C60).64C22, C30.4Q811C2, C30.160(C22×C4), (C2×C30).288C23, C22.16(C22×D15), (C2×Dic15).162C22, (C22×D15).126C22, C54(S3×C4⋊C4), C33(D5×C4⋊C4), C1514(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

C1C30 — C4⋊C4×D15
C1C5C15C30C2×C30C22×D15C2×C4×D15 — C4⋊C4×D15
C15C30 — C4⋊C4×D15
C1C22C4⋊C4

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 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×4], C10 [×3], Dic3 [×4], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], Dic5 [×4], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4×S3 [×8], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C22×S3, D15 [×4], C30 [×3], C2×C4⋊C4, C4×D5 [×8], C2×Dic5 [×3], C2×C20, C2×C20 [×2], C22×D5, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4 [×3], Dic15 [×2], Dic15 [×2], C60 [×2], C60 [×2], D30 [×6], C2×C30, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C2×C4×D5 [×3], S3×C4⋊C4, C4×D15 [×4], C4×D15 [×4], C2×Dic15, C2×Dic15 [×2], C2×C60, C2×C60 [×2], C22×D15, D5×C4⋊C4, C30.4Q8 [×2], C605C4, C15×C4⋊C4, C2×C4×D15, C2×C4×D15 [×2], C4⋊C4×D15
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C4×S3 [×2], C22×S3, D15, C2×C4⋊C4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4, S3×Q8, D30 [×3], C2×C4×D5, D4×D5, Q8×D5, S3×C4⋊C4, C4×D15 [×2], C22×D15, D5×C4⋊C4, C2×C4×D15, D4×D15, Q8×D15, C4⋊C4×D15

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

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4F4G···4L5A5B6A6B6C10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222222234···44···45566610···1012···121515151520···2030···3060···60
size11111515151522···230···30222222···24···422224···42···24···4

90 irreducible representations

dim11111122222222222444444
type+++++++-++++++-+-+-
imageC1C2C2C2C2C4S3D4Q8D5D6D10C4×S3D15C4×D5D30C4×D15S3×D4S3×Q8D4×D5Q8×D5D4×D15Q8×D15
kernelC4⋊C4×D15C30.4Q8C605C4C15×C4⋊C4C2×C4×D15C4×D15C5×C4⋊C4D30D30C3×C4⋊C4C2×C20C2×C12C20C4⋊C4C12C2×C4C4C10C10C6C6C2C2
# reps1211381222364481216112244

Matrix representation of C4⋊C4×D15 in GL4(𝔽61) generated by

60000
06000
005119
003010
,
11000
01100
00159
00160
,
473000
312500
0010
0001
,
333700
252800
00600
00060
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

׿
×
𝔽