direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×M5(2), C80⋊7C6, C4.C120, C48⋊7C10, C16⋊3C30, C8.2C60, C240⋊15C2, C24.6C20, C12.4C40, C60.16C8, C20.7C24, C22.C120, C40.11C12, C120.24C4, C120.114C22, (C2×C4).5C60, (C2×C6).1C40, (C2×C8).8C30, (C2×C30).5C8, C8.8(C2×C30), (C2×C60).54C4, (C2×C40).18C6, C4.12(C2×C60), C40.30(C2×C6), C6.13(C2×C40), (C2×C10).3C24, C2.3(C2×C120), C30.75(C2×C8), C10.22(C2×C24), C60.262(C2×C4), (C2×C20).25C12, C20.70(C2×C12), (C2×C120).38C2, C24.30(C2×C10), (C2×C12).14C20, (C2×C24).18C10, C12.49(C2×C20), SmallGroup(480,213)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×M5(2)
G = < a,b,c | a15=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Smallest permutation representation of C15×M5(2)
►On 240 points
Generators in S240
(1 108 61 180 72 30 232 198 45 119 175 142 213 94 150)(2 109 62 181 73 31 233 199 46 120 176 143 214 95 151)(3 110 63 182 74 32 234 200 47 121 161 144 215 96 152)(4 111 64 183 75 17 235 201 48 122 162 129 216 81 153)(5 112 49 184 76 18 236 202 33 123 163 130 217 82 154)(6 97 50 185 77 19 237 203 34 124 164 131 218 83 155)(7 98 51 186 78 20 238 204 35 125 165 132 219 84 156)(8 99 52 187 79 21 239 205 36 126 166 133 220 85 157)(9 100 53 188 80 22 240 206 37 127 167 134 221 86 158)(10 101 54 189 65 23 225 207 38 128 168 135 222 87 159)(11 102 55 190 66 24 226 208 39 113 169 136 223 88 160)(12 103 56 191 67 25 227 193 40 114 170 137 224 89 145)(13 104 57 192 68 26 228 194 41 115 171 138 209 90 146)(14 105 58 177 69 27 229 195 42 116 172 139 210 91 147)(15 106 59 178 70 28 230 196 43 117 173 140 211 92 148)(16 107 60 179 71 29 231 197 44 118 174 141 212 93 149)
(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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(97 105)(99 107)(101 109)(103 111)(114 122)(116 124)(118 126)(120 128)(129 137)(131 139)(133 141)(135 143)(145 153)(147 155)(149 157)(151 159)(162 170)(164 172)(166 174)(168 176)(177 185)(179 187)(181 189)(183 191)(193 201)(195 203)(197 205)(199 207)(210 218)(212 220)(214 222)(216 224)(225 233)(227 235)(229 237)(231 239)
G:=sub<Sym(240)| (1,108,61,180,72,30,232,198,45,119,175,142,213,94,150)(2,109,62,181,73,31,233,199,46,120,176,143,214,95,151)(3,110,63,182,74,32,234,200,47,121,161,144,215,96,152)(4,111,64,183,75,17,235,201,48,122,162,129,216,81,153)(5,112,49,184,76,18,236,202,33,123,163,130,217,82,154)(6,97,50,185,77,19,237,203,34,124,164,131,218,83,155)(7,98,51,186,78,20,238,204,35,125,165,132,219,84,156)(8,99,52,187,79,21,239,205,36,126,166,133,220,85,157)(9,100,53,188,80,22,240,206,37,127,167,134,221,86,158)(10,101,54,189,65,23,225,207,38,128,168,135,222,87,159)(11,102,55,190,66,24,226,208,39,113,169,136,223,88,160)(12,103,56,191,67,25,227,193,40,114,170,137,224,89,145)(13,104,57,192,68,26,228,194,41,115,171,138,209,90,146)(14,105,58,177,69,27,229,195,42,116,172,139,210,91,147)(15,106,59,178,70,28,230,196,43,117,173,140,211,92,148)(16,107,60,179,71,29,231,197,44,118,174,141,212,93,149), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(97,105)(99,107)(101,109)(103,111)(114,122)(116,124)(118,126)(120,128)(129,137)(131,139)(133,141)(135,143)(145,153)(147,155)(149,157)(151,159)(162,170)(164,172)(166,174)(168,176)(177,185)(179,187)(181,189)(183,191)(193,201)(195,203)(197,205)(199,207)(210,218)(212,220)(214,222)(216,224)(225,233)(227,235)(229,237)(231,239)>;
G:=Group( (1,108,61,180,72,30,232,198,45,119,175,142,213,94,150)(2,109,62,181,73,31,233,199,46,120,176,143,214,95,151)(3,110,63,182,74,32,234,200,47,121,161,144,215,96,152)(4,111,64,183,75,17,235,201,48,122,162,129,216,81,153)(5,112,49,184,76,18,236,202,33,123,163,130,217,82,154)(6,97,50,185,77,19,237,203,34,124,164,131,218,83,155)(7,98,51,186,78,20,238,204,35,125,165,132,219,84,156)(8,99,52,187,79,21,239,205,36,126,166,133,220,85,157)(9,100,53,188,80,22,240,206,37,127,167,134,221,86,158)(10,101,54,189,65,23,225,207,38,128,168,135,222,87,159)(11,102,55,190,66,24,226,208,39,113,169,136,223,88,160)(12,103,56,191,67,25,227,193,40,114,170,137,224,89,145)(13,104,57,192,68,26,228,194,41,115,171,138,209,90,146)(14,105,58,177,69,27,229,195,42,116,172,139,210,91,147)(15,106,59,178,70,28,230,196,43,117,173,140,211,92,148)(16,107,60,179,71,29,231,197,44,118,174,141,212,93,149), (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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31)(34,42)(36,44)(38,46)(40,48)(50,58)(52,60)(54,62)(56,64)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(97,105)(99,107)(101,109)(103,111)(114,122)(116,124)(118,126)(120,128)(129,137)(131,139)(133,141)(135,143)(145,153)(147,155)(149,157)(151,159)(162,170)(164,172)(166,174)(168,176)(177,185)(179,187)(181,189)(183,191)(193,201)(195,203)(197,205)(199,207)(210,218)(212,220)(214,222)(216,224)(225,233)(227,235)(229,237)(231,239) );
G=PermutationGroup([[(1,108,61,180,72,30,232,198,45,119,175,142,213,94,150),(2,109,62,181,73,31,233,199,46,120,176,143,214,95,151),(3,110,63,182,74,32,234,200,47,121,161,144,215,96,152),(4,111,64,183,75,17,235,201,48,122,162,129,216,81,153),(5,112,49,184,76,18,236,202,33,123,163,130,217,82,154),(6,97,50,185,77,19,237,203,34,124,164,131,218,83,155),(7,98,51,186,78,20,238,204,35,125,165,132,219,84,156),(8,99,52,187,79,21,239,205,36,126,166,133,220,85,157),(9,100,53,188,80,22,240,206,37,127,167,134,221,86,158),(10,101,54,189,65,23,225,207,38,128,168,135,222,87,159),(11,102,55,190,66,24,226,208,39,113,169,136,223,88,160),(12,103,56,191,67,25,227,193,40,114,170,137,224,89,145),(13,104,57,192,68,26,228,194,41,115,171,138,209,90,146),(14,105,58,177,69,27,229,195,42,116,172,139,210,91,147),(15,106,59,178,70,28,230,196,43,117,173,140,211,92,148),(16,107,60,179,71,29,231,197,44,118,174,141,212,93,149)], [(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)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31),(34,42),(36,44),(38,46),(40,48),(50,58),(52,60),(54,62),(56,64),(65,73),(67,75),(69,77),(71,79),(81,89),(83,91),(85,93),(87,95),(97,105),(99,107),(101,109),(103,111),(114,122),(116,124),(118,126),(120,128),(129,137),(131,139),(133,141),(135,143),(145,153),(147,155),(149,157),(151,159),(162,170),(164,172),(166,174),(168,176),(177,185),(179,187),(181,189),(183,191),(193,201),(195,203),(197,205),(199,207),(210,218),(212,220),(214,222),(216,224),(225,233),(227,235),(229,237),(231,239)]])
300 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 15A | ··· | 15H | 16A | ··· | 16H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 30A | ··· | 30H | 30I | ··· | 30P | 40A | ··· | 40P | 40Q | ··· | 40X | 48A | ··· | 48P | 60A | ··· | 60P | 60Q | ··· | 60X | 80A | ··· | 80AF | 120A | ··· | 120AF | 120AG | ··· | 120AV | 240A | ··· | 240BL |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C5 | C6 | C6 | C8 | C8 | C10 | C10 | C12 | C12 | C15 | C20 | C20 | C24 | C24 | C30 | C30 | C40 | C40 | C60 | C60 | C120 | C120 | M5(2) | C3×M5(2) | C5×M5(2) | C15×M5(2) |
| kernel | C15×M5(2) | C240 | C2×C120 | C5×M5(2) | C120 | C2×C60 | C3×M5(2) | C80 | C2×C40 | C60 | C2×C30 | C48 | C2×C24 | C40 | C2×C20 | M5(2) | C24 | C2×C12 | C20 | C2×C10 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 8 | 16 | 16 | 16 | 16 | 32 | 32 | 4 | 8 | 16 | 32 |
Matrix representation of C15×M5(2) ►in GL2(𝔽241) generated by
| 54 | 0 |
| 0 | 54 |
| 135 | 239 |
| 71 | 106 |
| 1 | 0 |
| 135 | 240 |
G:=sub<GL(2,GF(241))| [54,0,0,54],[135,71,239,106],[1,135,0,240] >;
C15×M5(2) in GAP, Magma, Sage, TeX
C_{15}\times M_5(2) % in TeX
G:=Group("C15xM5(2)"); // GroupNames label
G:=SmallGroup(480,213);
// by ID
G=gap.SmallGroup(480,213);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,420,3389,102,124]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations
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