Copied to
clipboard

## G = C15⋊Q16order 240 = 24·3·5

### 1st semidirect product of C15 and Q16 acting via Q16/C4=C22

Aliases: C151Q16, C20.7D6, C30.10D4, C12.7D10, Dic6.2D5, C60.25C22, Dic10.2S3, C4.18(S3×D5), C32(C5⋊Q16), C52(C3⋊Q16), C153C8.2C2, C6.10(C5⋊D4), C2.7(C15⋊D4), (C5×Dic6).2C2, C10.10(C3⋊D4), (C3×Dic10).2C2, SmallGroup(240,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C15⋊Q16
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — C15⋊Q16
 Lower central C15 — C30 — C60 — C15⋊Q16
 Upper central C1 — C2 — C4

Generators and relations for C15⋊Q16
G = < a,b,c | a15=b8=1, c2=b4, bab-1=a-1, cac-1=a11, cbc-1=b-1 >

Character table of C15⋊Q16

 class 1 2 3 4A 4B 4C 5A 5B 6 8A 8B 10A 10B 12A 12B 12C 15A 15B 20A 20B 20C 20D 20E 20F 30A 30B 60A 60B 60C 60D size 1 1 2 2 12 20 2 2 2 30 30 2 2 4 20 20 4 4 4 4 12 12 12 12 4 4 4 4 4 4 ρ1 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 1 1 trivial ρ2 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 1 1 linear of order 2 ρ3 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 1 1 linear of order 2 ρ4 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 1 1 linear of order 2 ρ5 2 2 2 -2 0 0 2 2 2 0 0 2 2 -2 0 0 2 2 -2 -2 0 0 0 0 2 2 -2 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 -1 2 0 2 2 2 -1 0 0 2 2 -1 -1 -1 -1 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 -1 2 0 -2 2 2 -1 0 0 2 2 -1 1 1 -1 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ8 2 2 2 2 -2 0 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ9 2 2 2 2 -2 0 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ10 2 2 2 2 2 0 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 2 2 2 2 2 0 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ12 2 -2 2 0 0 0 2 2 -2 -√2 √2 -2 -2 0 0 0 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 0 0 0 2 2 -2 √2 -√2 -2 -2 0 0 0 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ14 2 2 -1 -2 0 0 2 2 -1 0 0 2 2 1 -√-3 √-3 -1 -1 -2 -2 0 0 0 0 -1 -1 1 1 1 1 complex lifted from C3⋊D4 ρ15 2 2 2 -2 0 0 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 -2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ16 2 2 -1 -2 0 0 2 2 -1 0 0 2 2 1 √-3 -√-3 -1 -1 -2 -2 0 0 0 0 -1 -1 1 1 1 1 complex lifted from C3⋊D4 ρ17 2 2 2 -2 0 0 -1+√5/2 -1-√5/2 2 0 0 -1-√5/2 -1+√5/2 -2 0 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ18 2 2 2 -2 0 0 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 -2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ19 2 2 2 -2 0 0 -1-√5/2 -1+√5/2 2 0 0 -1+√5/2 -1-√5/2 -2 0 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ20 4 4 -2 4 0 0 -1+√5 -1-√5 -2 0 0 -1-√5 -1+√5 -2 0 0 1-√5/2 1+√5/2 -1+√5 -1-√5 0 0 0 0 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ21 4 4 -2 4 0 0 -1-√5 -1+√5 -2 0 0 -1+√5 -1-√5 -2 0 0 1+√5/2 1-√5/2 -1-√5 -1+√5 0 0 0 0 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ22 4 -4 -2 0 0 0 4 4 2 0 0 -4 -4 0 0 0 -2 -2 0 0 0 0 0 0 2 2 0 0 0 0 symplectic lifted from C3⋊Q16, Schur index 2 ρ23 4 4 -2 -4 0 0 -1-√5 -1+√5 -2 0 0 -1+√5 -1-√5 2 0 0 1+√5/2 1-√5/2 1+√5 1-√5 0 0 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 symplectic lifted from C15⋊D4, Schur index 2 ρ24 4 -4 4 0 0 0 -1+√5 -1-√5 -4 0 0 1+√5 1-√5 0 0 0 -1+√5 -1-√5 0 0 0 0 0 0 1-√5 1+√5 0 0 0 0 symplectic lifted from C5⋊Q16, Schur index 2 ρ25 4 4 -2 -4 0 0 -1+√5 -1-√5 -2 0 0 -1-√5 -1+√5 2 0 0 1-√5/2 1+√5/2 1-√5 1+√5 0 0 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 symplectic lifted from C15⋊D4, Schur index 2 ρ26 4 -4 4 0 0 0 -1-√5 -1+√5 -4 0 0 1-√5 1+√5 0 0 0 -1-√5 -1+√5 0 0 0 0 0 0 1+√5 1-√5 0 0 0 0 symplectic lifted from C5⋊Q16, Schur index 2 ρ27 4 -4 -2 0 0 0 -1+√5 -1-√5 2 0 0 1+√5 1-√5 0 0 0 1-√5/2 1+√5/2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -2ζ43ζ3ζ54+2ζ43ζ3ζ5-ζ43ζ54+ζ43ζ5 2ζ43ζ3ζ53-2ζ43ζ3ζ52+ζ43ζ53-ζ43ζ52 2ζ4ζ3ζ53-2ζ4ζ3ζ52+ζ4ζ53-ζ4ζ52 -2ζ4ζ3ζ54+2ζ4ζ3ζ5-ζ4ζ54+ζ4ζ5 complex faithful ρ28 4 -4 -2 0 0 0 -1-√5 -1+√5 2 0 0 1-√5 1+√5 0 0 0 1+√5/2 1-√5/2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 2ζ43ζ3ζ53-2ζ43ζ3ζ52+ζ43ζ53-ζ43ζ52 -2ζ4ζ3ζ54+2ζ4ζ3ζ5-ζ4ζ54+ζ4ζ5 -2ζ43ζ3ζ54+2ζ43ζ3ζ5-ζ43ζ54+ζ43ζ5 2ζ4ζ3ζ53-2ζ4ζ3ζ52+ζ4ζ53-ζ4ζ52 complex faithful ρ29 4 -4 -2 0 0 0 -1+√5 -1-√5 2 0 0 1+√5 1-√5 0 0 0 1-√5/2 1+√5/2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -2ζ4ζ3ζ54+2ζ4ζ3ζ5-ζ4ζ54+ζ4ζ5 2ζ4ζ3ζ53-2ζ4ζ3ζ52+ζ4ζ53-ζ4ζ52 2ζ43ζ3ζ53-2ζ43ζ3ζ52+ζ43ζ53-ζ43ζ52 -2ζ43ζ3ζ54+2ζ43ζ3ζ5-ζ43ζ54+ζ43ζ5 complex faithful ρ30 4 -4 -2 0 0 0 -1-√5 -1+√5 2 0 0 1-√5 1+√5 0 0 0 1+√5/2 1-√5/2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 2ζ4ζ3ζ53-2ζ4ζ3ζ52+ζ4ζ53-ζ4ζ52 -2ζ43ζ3ζ54+2ζ43ζ3ζ5-ζ43ζ54+ζ43ζ5 -2ζ4ζ3ζ54+2ζ4ζ3ζ5-ζ4ζ54+ζ4ζ5 2ζ43ζ3ζ53-2ζ43ζ3ζ52+ζ43ζ53-ζ43ζ52 complex faithful

Smallest permutation representation of C15⋊Q16
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)
(1 211 35 200 16 228 54 192)(2 225 36 199 17 227 55 191)(3 224 37 198 18 226 56 190)(4 223 38 197 19 240 57 189)(5 222 39 196 20 239 58 188)(6 221 40 210 21 238 59 187)(7 220 41 209 22 237 60 186)(8 219 42 208 23 236 46 185)(9 218 43 207 24 235 47 184)(10 217 44 206 25 234 48 183)(11 216 45 205 26 233 49 182)(12 215 31 204 27 232 50 181)(13 214 32 203 28 231 51 195)(14 213 33 202 29 230 52 194)(15 212 34 201 30 229 53 193)(61 164 108 132 86 175 100 138)(62 163 109 131 87 174 101 137)(63 162 110 130 88 173 102 136)(64 161 111 129 89 172 103 150)(65 160 112 128 90 171 104 149)(66 159 113 127 76 170 105 148)(67 158 114 126 77 169 91 147)(68 157 115 125 78 168 92 146)(69 156 116 124 79 167 93 145)(70 155 117 123 80 166 94 144)(71 154 118 122 81 180 95 143)(72 153 119 121 82 179 96 142)(73 152 120 135 83 178 97 141)(74 151 106 134 84 177 98 140)(75 165 107 133 85 176 99 139)
(1 88 16 63)(2 84 17 74)(3 80 18 70)(4 76 19 66)(5 87 20 62)(6 83 21 73)(7 79 22 69)(8 90 23 65)(9 86 24 61)(10 82 25 72)(11 78 26 68)(12 89 27 64)(13 85 28 75)(14 81 29 71)(15 77 30 67)(31 111 50 103)(32 107 51 99)(33 118 52 95)(34 114 53 91)(35 110 54 102)(36 106 55 98)(37 117 56 94)(38 113 57 105)(39 109 58 101)(40 120 59 97)(41 116 60 93)(42 112 46 104)(43 108 47 100)(44 119 48 96)(45 115 49 92)(121 234 142 217)(122 230 143 213)(123 226 144 224)(124 237 145 220)(125 233 146 216)(126 229 147 212)(127 240 148 223)(128 236 149 219)(129 232 150 215)(130 228 136 211)(131 239 137 222)(132 235 138 218)(133 231 139 214)(134 227 140 225)(135 238 141 221)(151 191 177 199)(152 187 178 210)(153 183 179 206)(154 194 180 202)(155 190 166 198)(156 186 167 209)(157 182 168 205)(158 193 169 201)(159 189 170 197)(160 185 171 208)(161 181 172 204)(162 192 173 200)(163 188 174 196)(164 184 175 207)(165 195 176 203)```

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

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

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

C15⋊Q16 is a maximal subgroup of
Dic10.D6  Dic6.D10  D30.3D4  D30.4D4  D20.34D6  D20.37D6  D12.37D10  C60.8C23  C60.10C23  C60.16C23  C60.19C23  D5×C3⋊Q16  S3×C5⋊Q16  C60.39C23  C60.44C23
C15⋊Q16 is a maximal quotient of
C30.Q16  Dic6⋊Dic5  C30.20D8

Matrix representation of C15⋊Q16 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 190 51 0 0 0 0 190 240 0 0 0 0 0 0 239 192 0 0 0 0 64 1
,
 0 219 0 0 0 0 11 219 0 0 0 0 0 0 131 80 0 0 0 0 147 110 0 0 0 0 0 0 131 98 0 0 0 0 179 110
,
 25 83 0 0 0 0 187 216 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 160 117 0 0 0 0 84 81

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,190,190,0,0,0,0,51,240,0,0,0,0,0,0,239,64,0,0,0,0,192,1],[0,11,0,0,0,0,219,219,0,0,0,0,0,0,131,147,0,0,0,0,80,110,0,0,0,0,0,0,131,179,0,0,0,0,98,110],[25,187,0,0,0,0,83,216,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,160,84,0,0,0,0,117,81] >;`

C15⋊Q16 in GAP, Magma, Sage, TeX

`C_{15}\rtimes Q_{16}`
`% in TeX`

`G:=Group("C15:Q16");`
`// GroupNames label`

`G:=SmallGroup(240,22);`
`// by ID`

`G=gap.SmallGroup(240,22);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,55,218,116,50,490,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^15=b^8=1,c^2=b^4,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=b^-1>;`
`// generators/relations`

Export

׿
×
𝔽