direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C15⋊3C8, C30⋊3C8, C60.9C4, C20.49D6, C4.14D30, C12.50D10, C20.6Dic3, C12.3Dic5, C4.3Dic15, C60.56C22, C22.2Dic15, C6⋊(C5⋊2C8), C10⋊2(C3⋊C8), C15⋊13(C2×C8), (C2×C60).8C2, (C2×C30).6C4, (C2×C20).6S3, (C2×C12).6D5, (C2×C4).5D15, C30.50(C2×C4), (C2×C6).2Dic5, C6.6(C2×Dic5), C2.1(C2×Dic15), (C2×C10).4Dic3, C10.13(C2×Dic3), C5⋊4(C2×C3⋊C8), C3⋊2(C2×C5⋊2C8), SmallGroup(240,70)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C2×C15⋊3C8 |
Generators and relations for C2×C15⋊3C8
G = < a,b,c | a2=b15=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C15⋊3C8
►Regular action on 240 points
Generators in S240
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 61)(16 86)(17 87)(18 88)(19 89)(20 90)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 105)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 117)(47 118)(48 119)(49 120)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(127 194)(128 195)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 196)(137 197)(138 198)(139 199)(140 200)(141 201)(142 202)(143 203)(144 204)(145 205)(146 206)(147 207)(148 208)(149 209)(150 210)(151 214)(152 215)(153 216)(154 217)(155 218)(156 219)(157 220)(158 221)(159 222)(160 223)(161 224)(162 225)(163 211)(164 212)(165 213)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)
(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)
(1 238 56 204 20 215 45 181)(2 237 57 203 21 214 31 195)(3 236 58 202 22 213 32 194)(4 235 59 201 23 212 33 193)(5 234 60 200 24 211 34 192)(6 233 46 199 25 225 35 191)(7 232 47 198 26 224 36 190)(8 231 48 197 27 223 37 189)(9 230 49 196 28 222 38 188)(10 229 50 210 29 221 39 187)(11 228 51 209 30 220 40 186)(12 227 52 208 16 219 41 185)(13 226 53 207 17 218 42 184)(14 240 54 206 18 217 43 183)(15 239 55 205 19 216 44 182)(61 179 111 145 89 153 103 130)(62 178 112 144 90 152 104 129)(63 177 113 143 76 151 105 128)(64 176 114 142 77 165 91 127)(65 175 115 141 78 164 92 126)(66 174 116 140 79 163 93 125)(67 173 117 139 80 162 94 124)(68 172 118 138 81 161 95 123)(69 171 119 137 82 160 96 122)(70 170 120 136 83 159 97 121)(71 169 106 150 84 158 98 135)(72 168 107 149 85 157 99 134)(73 167 108 148 86 156 100 133)(74 166 109 147 87 155 101 132)(75 180 110 146 88 154 102 131)
G:=sub<Sym(240)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,61)(16,86)(17,87)(18,88)(19,89)(20,90)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,105)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,117)(47,118)(48,119)(49,120)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,196)(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,214)(152,215)(153,216)(154,217)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)(161,224)(162,225)(163,211)(164,212)(165,213)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240), (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), (1,238,56,204,20,215,45,181)(2,237,57,203,21,214,31,195)(3,236,58,202,22,213,32,194)(4,235,59,201,23,212,33,193)(5,234,60,200,24,211,34,192)(6,233,46,199,25,225,35,191)(7,232,47,198,26,224,36,190)(8,231,48,197,27,223,37,189)(9,230,49,196,28,222,38,188)(10,229,50,210,29,221,39,187)(11,228,51,209,30,220,40,186)(12,227,52,208,16,219,41,185)(13,226,53,207,17,218,42,184)(14,240,54,206,18,217,43,183)(15,239,55,205,19,216,44,182)(61,179,111,145,89,153,103,130)(62,178,112,144,90,152,104,129)(63,177,113,143,76,151,105,128)(64,176,114,142,77,165,91,127)(65,175,115,141,78,164,92,126)(66,174,116,140,79,163,93,125)(67,173,117,139,80,162,94,124)(68,172,118,138,81,161,95,123)(69,171,119,137,82,160,96,122)(70,170,120,136,83,159,97,121)(71,169,106,150,84,158,98,135)(72,168,107,149,85,157,99,134)(73,167,108,148,86,156,100,133)(74,166,109,147,87,155,101,132)(75,180,110,146,88,154,102,131)>;
G:=Group( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,61)(16,86)(17,87)(18,88)(19,89)(20,90)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,105)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,117)(47,118)(48,119)(49,120)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,196)(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,214)(152,215)(153,216)(154,217)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)(161,224)(162,225)(163,211)(164,212)(165,213)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240), (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), (1,238,56,204,20,215,45,181)(2,237,57,203,21,214,31,195)(3,236,58,202,22,213,32,194)(4,235,59,201,23,212,33,193)(5,234,60,200,24,211,34,192)(6,233,46,199,25,225,35,191)(7,232,47,198,26,224,36,190)(8,231,48,197,27,223,37,189)(9,230,49,196,28,222,38,188)(10,229,50,210,29,221,39,187)(11,228,51,209,30,220,40,186)(12,227,52,208,16,219,41,185)(13,226,53,207,17,218,42,184)(14,240,54,206,18,217,43,183)(15,239,55,205,19,216,44,182)(61,179,111,145,89,153,103,130)(62,178,112,144,90,152,104,129)(63,177,113,143,76,151,105,128)(64,176,114,142,77,165,91,127)(65,175,115,141,78,164,92,126)(66,174,116,140,79,163,93,125)(67,173,117,139,80,162,94,124)(68,172,118,138,81,161,95,123)(69,171,119,137,82,160,96,122)(70,170,120,136,83,159,97,121)(71,169,106,150,84,158,98,135)(72,168,107,149,85,157,99,134)(73,167,108,148,86,156,100,133)(74,166,109,147,87,155,101,132)(75,180,110,146,88,154,102,131) );
G=PermutationGroup([[(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,61),(16,86),(17,87),(18,88),(19,89),(20,90),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,105),(32,91),(33,92),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,99),(41,100),(42,101),(43,102),(44,103),(45,104),(46,117),(47,118),(48,119),(49,120),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(121,188),(122,189),(123,190),(124,191),(125,192),(126,193),(127,194),(128,195),(129,181),(130,182),(131,183),(132,184),(133,185),(134,186),(135,187),(136,196),(137,197),(138,198),(139,199),(140,200),(141,201),(142,202),(143,203),(144,204),(145,205),(146,206),(147,207),(148,208),(149,209),(150,210),(151,214),(152,215),(153,216),(154,217),(155,218),(156,219),(157,220),(158,221),(159,222),(160,223),(161,224),(162,225),(163,211),(164,212),(165,213),(166,226),(167,227),(168,228),(169,229),(170,230),(171,231),(172,232),(173,233),(174,234),(175,235),(176,236),(177,237),(178,238),(179,239),(180,240)], [(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)], [(1,238,56,204,20,215,45,181),(2,237,57,203,21,214,31,195),(3,236,58,202,22,213,32,194),(4,235,59,201,23,212,33,193),(5,234,60,200,24,211,34,192),(6,233,46,199,25,225,35,191),(7,232,47,198,26,224,36,190),(8,231,48,197,27,223,37,189),(9,230,49,196,28,222,38,188),(10,229,50,210,29,221,39,187),(11,228,51,209,30,220,40,186),(12,227,52,208,16,219,41,185),(13,226,53,207,17,218,42,184),(14,240,54,206,18,217,43,183),(15,239,55,205,19,216,44,182),(61,179,111,145,89,153,103,130),(62,178,112,144,90,152,104,129),(63,177,113,143,76,151,105,128),(64,176,114,142,77,165,91,127),(65,175,115,141,78,164,92,126),(66,174,116,140,79,163,93,125),(67,173,117,139,80,162,94,124),(68,172,118,138,81,161,95,123),(69,171,119,137,82,160,96,122),(70,170,120,136,83,159,97,121),(71,169,106,150,84,158,98,135),(72,168,107,149,85,157,99,134),(73,167,108,148,86,156,100,133),(74,166,109,147,87,155,101,132),(75,180,110,146,88,154,102,131)]])
C2×C15⋊3C8 is a maximal subgroup of
Dic5×C3⋊C8 Dic3×C5⋊2C8 C30.21C42 C30.22C42 C60.93D4 C60.94D4 C30.D8 D12⋊Dic5 C30.Q16 Dic6⋊Dic5 C60.13Q8 C60.15Q8 C30.SD16 C30.20D8 C60.D4 C42.D15 C60⋊5C8 C60.1Q8 C60.2Q8 D60⋊9C4 Dic30⋊9C4 C8×Dic15 C60.26Q8 C120⋊13C4 D30⋊3C8 C60.210D4 C60.212D4 D4⋊Dic15 Q8⋊2Dic15 C2×D5×C3⋊C8 D20.2Dic3 C2×S3×C5⋊2C8 D12.Dic5 D20.34D6 C2×C8×D15 D60.3C4 D4.Dic15 D4.8D30
C2×C15⋊3C8 is a maximal quotient of
C60⋊5C8 C60.7C8 C60.212D4
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | ··· | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 15 | ··· | 15 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | - | + | - | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D5 | Dic3 | D6 | Dic3 | Dic5 | D10 | Dic5 | C3⋊C8 | D15 | C5⋊2C8 | Dic15 | D30 | Dic15 | C15⋊3C8 |
| kernel | C2×C15⋊3C8 | C15⋊3C8 | C2×C60 | C60 | C2×C30 | C30 | C2×C20 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 16 |
Matrix representation of C2×C15⋊3C8 ►in GL5(𝔽241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 52 | 52 | 0 | 0 |
| 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 225 | 131 |
| 0 | 0 | 0 | 46 | 30 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 35 | 2 | 0 | 0 |
| 0 | 110 | 206 | 0 | 0 |
| 0 | 0 | 0 | 184 | 177 |
| 0 | 0 | 0 | 111 | 57 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,52,189,0,0,0,52,240,0,0,0,0,0,225,46,0,0,0,131,30],[8,0,0,0,0,0,35,110,0,0,0,2,206,0,0,0,0,0,184,111,0,0,0,177,57] >;
C2×C15⋊3C8 in GAP, Magma, Sage, TeX
C_2\times C_{15}\rtimes_3C_8 % in TeX
G:=Group("C2xC15:3C8"); // GroupNames label
G:=SmallGroup(240,70);
// by ID
G=gap.SmallGroup(240,70);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,50,964,6917]);
// Polycyclic
G:=Group<a,b,c|a^2=b^15=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export