metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊7Q16, C4.4D30, C30.37D4, C20.12D6, Q8.2D15, C12.12D10, C60.4C22, Dic30.2C2, C5⋊3(C3⋊Q16), C3⋊3(C5⋊Q16), (C5×Q8).3S3, (C3×Q8).1D5, C15⋊3C8.1C2, (Q8×C15).1C2, C6.19(C5⋊D4), C2.7(C15⋊7D4), C10.19(C3⋊D4), SmallGroup(240,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊7Q16
G = < a,b,c | a15=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C15⋊7Q16
►Regular action on 240 points
Generators in S240
(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)
(1 217 35 210 20 240 50 195)(2 216 36 209 21 239 51 194)(3 215 37 208 22 238 52 193)(4 214 38 207 23 237 53 192)(5 213 39 206 24 236 54 191)(6 212 40 205 25 235 55 190)(7 211 41 204 26 234 56 189)(8 225 42 203 27 233 57 188)(9 224 43 202 28 232 58 187)(10 223 44 201 29 231 59 186)(11 222 45 200 30 230 60 185)(12 221 31 199 16 229 46 184)(13 220 32 198 17 228 47 183)(14 219 33 197 18 227 48 182)(15 218 34 196 19 226 49 181)(61 152 117 125 76 176 95 140)(62 151 118 124 77 175 96 139)(63 165 119 123 78 174 97 138)(64 164 120 122 79 173 98 137)(65 163 106 121 80 172 99 136)(66 162 107 135 81 171 100 150)(67 161 108 134 82 170 101 149)(68 160 109 133 83 169 102 148)(69 159 110 132 84 168 103 147)(70 158 111 131 85 167 104 146)(71 157 112 130 86 166 105 145)(72 156 113 129 87 180 91 144)(73 155 114 128 88 179 92 143)(74 154 115 127 89 178 93 142)(75 153 116 126 90 177 94 141)
(1 84 20 69)(2 85 21 70)(3 86 22 71)(4 87 23 72)(5 88 24 73)(6 89 25 74)(7 90 26 75)(8 76 27 61)(9 77 28 62)(10 78 29 63)(11 79 30 64)(12 80 16 65)(13 81 17 66)(14 82 18 67)(15 83 19 68)(31 106 46 99)(32 107 47 100)(33 108 48 101)(34 109 49 102)(35 110 50 103)(36 111 51 104)(37 112 52 105)(38 113 53 91)(39 114 54 92)(40 115 55 93)(41 116 56 94)(42 117 57 95)(43 118 58 96)(44 119 59 97)(45 120 60 98)(121 229 136 221)(122 230 137 222)(123 231 138 223)(124 232 139 224)(125 233 140 225)(126 234 141 211)(127 235 142 212)(128 236 143 213)(129 237 144 214)(130 238 145 215)(131 239 146 216)(132 240 147 217)(133 226 148 218)(134 227 149 219)(135 228 150 220)(151 187 175 202)(152 188 176 203)(153 189 177 204)(154 190 178 205)(155 191 179 206)(156 192 180 207)(157 193 166 208)(158 194 167 209)(159 195 168 210)(160 181 169 196)(161 182 170 197)(162 183 171 198)(163 184 172 199)(164 185 173 200)(165 186 174 201)
G:=sub<Sym(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), (1,217,35,210,20,240,50,195)(2,216,36,209,21,239,51,194)(3,215,37,208,22,238,52,193)(4,214,38,207,23,237,53,192)(5,213,39,206,24,236,54,191)(6,212,40,205,25,235,55,190)(7,211,41,204,26,234,56,189)(8,225,42,203,27,233,57,188)(9,224,43,202,28,232,58,187)(10,223,44,201,29,231,59,186)(11,222,45,200,30,230,60,185)(12,221,31,199,16,229,46,184)(13,220,32,198,17,228,47,183)(14,219,33,197,18,227,48,182)(15,218,34,196,19,226,49,181)(61,152,117,125,76,176,95,140)(62,151,118,124,77,175,96,139)(63,165,119,123,78,174,97,138)(64,164,120,122,79,173,98,137)(65,163,106,121,80,172,99,136)(66,162,107,135,81,171,100,150)(67,161,108,134,82,170,101,149)(68,160,109,133,83,169,102,148)(69,159,110,132,84,168,103,147)(70,158,111,131,85,167,104,146)(71,157,112,130,86,166,105,145)(72,156,113,129,87,180,91,144)(73,155,114,128,88,179,92,143)(74,154,115,127,89,178,93,142)(75,153,116,126,90,177,94,141), (1,84,20,69)(2,85,21,70)(3,86,22,71)(4,87,23,72)(5,88,24,73)(6,89,25,74)(7,90,26,75)(8,76,27,61)(9,77,28,62)(10,78,29,63)(11,79,30,64)(12,80,16,65)(13,81,17,66)(14,82,18,67)(15,83,19,68)(31,106,46,99)(32,107,47,100)(33,108,48,101)(34,109,49,102)(35,110,50,103)(36,111,51,104)(37,112,52,105)(38,113,53,91)(39,114,54,92)(40,115,55,93)(41,116,56,94)(42,117,57,95)(43,118,58,96)(44,119,59,97)(45,120,60,98)(121,229,136,221)(122,230,137,222)(123,231,138,223)(124,232,139,224)(125,233,140,225)(126,234,141,211)(127,235,142,212)(128,236,143,213)(129,237,144,214)(130,238,145,215)(131,239,146,216)(132,240,147,217)(133,226,148,218)(134,227,149,219)(135,228,150,220)(151,187,175,202)(152,188,176,203)(153,189,177,204)(154,190,178,205)(155,191,179,206)(156,192,180,207)(157,193,166,208)(158,194,167,209)(159,195,168,210)(160,181,169,196)(161,182,170,197)(162,183,171,198)(163,184,172,199)(164,185,173,200)(165,186,174,201)>;
G:=Group( (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), (1,217,35,210,20,240,50,195)(2,216,36,209,21,239,51,194)(3,215,37,208,22,238,52,193)(4,214,38,207,23,237,53,192)(5,213,39,206,24,236,54,191)(6,212,40,205,25,235,55,190)(7,211,41,204,26,234,56,189)(8,225,42,203,27,233,57,188)(9,224,43,202,28,232,58,187)(10,223,44,201,29,231,59,186)(11,222,45,200,30,230,60,185)(12,221,31,199,16,229,46,184)(13,220,32,198,17,228,47,183)(14,219,33,197,18,227,48,182)(15,218,34,196,19,226,49,181)(61,152,117,125,76,176,95,140)(62,151,118,124,77,175,96,139)(63,165,119,123,78,174,97,138)(64,164,120,122,79,173,98,137)(65,163,106,121,80,172,99,136)(66,162,107,135,81,171,100,150)(67,161,108,134,82,170,101,149)(68,160,109,133,83,169,102,148)(69,159,110,132,84,168,103,147)(70,158,111,131,85,167,104,146)(71,157,112,130,86,166,105,145)(72,156,113,129,87,180,91,144)(73,155,114,128,88,179,92,143)(74,154,115,127,89,178,93,142)(75,153,116,126,90,177,94,141), (1,84,20,69)(2,85,21,70)(3,86,22,71)(4,87,23,72)(5,88,24,73)(6,89,25,74)(7,90,26,75)(8,76,27,61)(9,77,28,62)(10,78,29,63)(11,79,30,64)(12,80,16,65)(13,81,17,66)(14,82,18,67)(15,83,19,68)(31,106,46,99)(32,107,47,100)(33,108,48,101)(34,109,49,102)(35,110,50,103)(36,111,51,104)(37,112,52,105)(38,113,53,91)(39,114,54,92)(40,115,55,93)(41,116,56,94)(42,117,57,95)(43,118,58,96)(44,119,59,97)(45,120,60,98)(121,229,136,221)(122,230,137,222)(123,231,138,223)(124,232,139,224)(125,233,140,225)(126,234,141,211)(127,235,142,212)(128,236,143,213)(129,237,144,214)(130,238,145,215)(131,239,146,216)(132,240,147,217)(133,226,148,218)(134,227,149,219)(135,228,150,220)(151,187,175,202)(152,188,176,203)(153,189,177,204)(154,190,178,205)(155,191,179,206)(156,192,180,207)(157,193,166,208)(158,194,167,209)(159,195,168,210)(160,181,169,196)(161,182,170,197)(162,183,171,198)(163,184,172,199)(164,185,173,200)(165,186,174,201) );
G=PermutationGroup([[(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)], [(1,217,35,210,20,240,50,195),(2,216,36,209,21,239,51,194),(3,215,37,208,22,238,52,193),(4,214,38,207,23,237,53,192),(5,213,39,206,24,236,54,191),(6,212,40,205,25,235,55,190),(7,211,41,204,26,234,56,189),(8,225,42,203,27,233,57,188),(9,224,43,202,28,232,58,187),(10,223,44,201,29,231,59,186),(11,222,45,200,30,230,60,185),(12,221,31,199,16,229,46,184),(13,220,32,198,17,228,47,183),(14,219,33,197,18,227,48,182),(15,218,34,196,19,226,49,181),(61,152,117,125,76,176,95,140),(62,151,118,124,77,175,96,139),(63,165,119,123,78,174,97,138),(64,164,120,122,79,173,98,137),(65,163,106,121,80,172,99,136),(66,162,107,135,81,171,100,150),(67,161,108,134,82,170,101,149),(68,160,109,133,83,169,102,148),(69,159,110,132,84,168,103,147),(70,158,111,131,85,167,104,146),(71,157,112,130,86,166,105,145),(72,156,113,129,87,180,91,144),(73,155,114,128,88,179,92,143),(74,154,115,127,89,178,93,142),(75,153,116,126,90,177,94,141)], [(1,84,20,69),(2,85,21,70),(3,86,22,71),(4,87,23,72),(5,88,24,73),(6,89,25,74),(7,90,26,75),(8,76,27,61),(9,77,28,62),(10,78,29,63),(11,79,30,64),(12,80,16,65),(13,81,17,66),(14,82,18,67),(15,83,19,68),(31,106,46,99),(32,107,47,100),(33,108,48,101),(34,109,49,102),(35,110,50,103),(36,111,51,104),(37,112,52,105),(38,113,53,91),(39,114,54,92),(40,115,55,93),(41,116,56,94),(42,117,57,95),(43,118,58,96),(44,119,59,97),(45,120,60,98),(121,229,136,221),(122,230,137,222),(123,231,138,223),(124,232,139,224),(125,233,140,225),(126,234,141,211),(127,235,142,212),(128,236,143,213),(129,237,144,214),(130,238,145,215),(131,239,146,216),(132,240,147,217),(133,226,148,218),(134,227,149,219),(135,228,150,220),(151,187,175,202),(152,188,176,203),(153,189,177,204),(154,190,178,205),(155,191,179,206),(156,192,180,207),(157,193,166,208),(158,194,167,209),(159,195,168,210),(160,181,169,196),(161,182,170,197),(162,183,171,198),(163,184,172,199),(164,185,173,200),(165,186,174,201)]])
C15⋊7Q16 is a maximal subgroup of
D5×C3⋊Q16 D20.13D6 S3×C5⋊Q16 Dic10.26D6 D12.27D10 D20.14D6 D20.27D6 D20.28D6 SD16⋊D15 D4.5D30 Q16×D15 Q16⋊D15 Q8.11D30 D4.8D30 D4.9D30
C15⋊7Q16 is a maximal quotient of
C60.1Q8 Dic30⋊9C4 Q8⋊2Dic15
42 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20F | 30A | 30B | 30C | 30D | 60A | ··· | 60L |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 4 | 60 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | Q16 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 | C3⋊Q16 | C5⋊Q16 | C15⋊7Q16 |
| kernel | C15⋊7Q16 | C15⋊3C8 | Dic30 | Q8×C15 | C5×Q8 | C30 | C3×Q8 | C20 | C15 | C12 | C10 | Q8 | C6 | C4 | C2 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 4 |
Matrix representation of C15⋊7Q16 ►in GL6(𝔽241)
| 225 | 144 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 51 | 0 | 0 |
| 0 | 0 | 189 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 122 | 35 | 0 | 0 | 0 | 0 |
| 43 | 119 | 0 | 0 | 0 | 0 |
| 0 | 0 | 204 | 108 | 0 | 0 |
| 0 | 0 | 148 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 230 | 11 |
| 0 | 0 | 0 | 0 | 230 | 230 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 128 |
| 0 | 0 | 0 | 0 | 128 | 185 |
G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,144,15,0,0,0,0,0,0,189,189,0,0,0,0,51,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[122,43,0,0,0,0,35,119,0,0,0,0,0,0,204,148,0,0,0,0,108,37,0,0,0,0,0,0,230,230,0,0,0,0,11,230],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,56,128,0,0,0,0,128,185] >;
C15⋊7Q16 in GAP, Magma, Sage, TeX
C_{15}\rtimes_7Q_{16} % in TeX
G:=Group("C15:7Q16"); // GroupNames label
G:=SmallGroup(240,79);
// by ID
G=gap.SmallGroup(240,79);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,55,218,116,50,964,6917]);
// Polycyclic
G:=Group<a,b,c|a^15=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export