metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.5D8, C40.7D6, C15⋊2SD32, D24.2D5, C60.74D4, C24.7D10, Dic20⋊3S3, C120.34C22, C3⋊2(D8.D5), C15⋊3C16⋊6C2, C8.27(S3×D5), (C5×D24).2C2, C5⋊2(C8.6D6), C6.10(D4⋊D5), (C3×Dic20)⋊6C2, C2.6(C15⋊D8), C12.3(C5⋊D4), C20.3(C3⋊D4), C4.3(C15⋊D4), C10.10(D4⋊S3), SmallGroup(480,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊SD32
G = < a,b,c | a15=b16=c2=1, bab-1=a-1, cac=a11, cbc=b7 >
Smallest permutation representation of C15⋊SD32
►On 240 points
Generators in S240
(1 202 118 220 103 29 137 185 148 234 90 45 55 161 68)(2 69 162 56 46 91 235 149 186 138 30 104 221 119 203)(3 204 120 222 105 31 139 187 150 236 92 47 57 163 70)(4 71 164 58 48 93 237 151 188 140 32 106 223 121 205)(5 206 122 224 107 17 141 189 152 238 94 33 59 165 72)(6 73 166 60 34 95 239 153 190 142 18 108 209 123 207)(7 208 124 210 109 19 143 191 154 240 96 35 61 167 74)(8 75 168 62 36 81 225 155 192 144 20 110 211 125 193)(9 194 126 212 111 21 129 177 156 226 82 37 63 169 76)(10 77 170 64 38 83 227 157 178 130 22 112 213 127 195)(11 196 128 214 97 23 131 179 158 228 84 39 49 171 78)(12 79 172 50 40 85 229 159 180 132 24 98 215 113 197)(13 198 114 216 99 25 133 181 160 230 86 41 51 173 80)(14 65 174 52 42 87 231 145 182 134 26 100 217 115 199)(15 200 116 218 101 27 135 183 146 232 88 43 53 175 66)(16 67 176 54 44 89 233 147 184 136 28 102 219 117 201)
(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 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 86)(18 93)(19 84)(20 91)(21 82)(22 89)(23 96)(24 87)(25 94)(26 85)(27 92)(28 83)(29 90)(30 81)(31 88)(32 95)(33 198)(34 205)(35 196)(36 203)(37 194)(38 201)(39 208)(40 199)(41 206)(42 197)(43 204)(44 195)(45 202)(46 193)(47 200)(48 207)(49 61)(50 52)(51 59)(53 57)(54 64)(56 62)(58 60)(65 98)(66 105)(67 112)(68 103)(69 110)(70 101)(71 108)(72 99)(73 106)(74 97)(75 104)(76 111)(77 102)(78 109)(79 100)(80 107)(113 182)(114 189)(115 180)(116 187)(117 178)(118 185)(119 192)(120 183)(121 190)(122 181)(123 188)(124 179)(125 186)(126 177)(127 184)(128 191)(130 136)(131 143)(132 134)(133 141)(135 139)(138 144)(140 142)(145 172)(146 163)(147 170)(148 161)(149 168)(150 175)(151 166)(152 173)(153 164)(154 171)(155 162)(156 169)(157 176)(158 167)(159 174)(160 165)(209 223)(210 214)(211 221)(213 219)(215 217)(216 224)(218 222)(225 235)(227 233)(228 240)(229 231)(230 238)(232 236)(237 239)
G:=sub<Sym(240)| (1,202,118,220,103,29,137,185,148,234,90,45,55,161,68)(2,69,162,56,46,91,235,149,186,138,30,104,221,119,203)(3,204,120,222,105,31,139,187,150,236,92,47,57,163,70)(4,71,164,58,48,93,237,151,188,140,32,106,223,121,205)(5,206,122,224,107,17,141,189,152,238,94,33,59,165,72)(6,73,166,60,34,95,239,153,190,142,18,108,209,123,207)(7,208,124,210,109,19,143,191,154,240,96,35,61,167,74)(8,75,168,62,36,81,225,155,192,144,20,110,211,125,193)(9,194,126,212,111,21,129,177,156,226,82,37,63,169,76)(10,77,170,64,38,83,227,157,178,130,22,112,213,127,195)(11,196,128,214,97,23,131,179,158,228,84,39,49,171,78)(12,79,172,50,40,85,229,159,180,132,24,98,215,113,197)(13,198,114,216,99,25,133,181,160,230,86,41,51,173,80)(14,65,174,52,42,87,231,145,182,134,26,100,217,115,199)(15,200,116,218,101,27,135,183,146,232,88,43,53,175,66)(16,67,176,54,44,89,233,147,184,136,28,102,219,117,201), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,86)(18,93)(19,84)(20,91)(21,82)(22,89)(23,96)(24,87)(25,94)(26,85)(27,92)(28,83)(29,90)(30,81)(31,88)(32,95)(33,198)(34,205)(35,196)(36,203)(37,194)(38,201)(39,208)(40,199)(41,206)(42,197)(43,204)(44,195)(45,202)(46,193)(47,200)(48,207)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(65,98)(66,105)(67,112)(68,103)(69,110)(70,101)(71,108)(72,99)(73,106)(74,97)(75,104)(76,111)(77,102)(78,109)(79,100)(80,107)(113,182)(114,189)(115,180)(116,187)(117,178)(118,185)(119,192)(120,183)(121,190)(122,181)(123,188)(124,179)(125,186)(126,177)(127,184)(128,191)(130,136)(131,143)(132,134)(133,141)(135,139)(138,144)(140,142)(145,172)(146,163)(147,170)(148,161)(149,168)(150,175)(151,166)(152,173)(153,164)(154,171)(155,162)(156,169)(157,176)(158,167)(159,174)(160,165)(209,223)(210,214)(211,221)(213,219)(215,217)(216,224)(218,222)(225,235)(227,233)(228,240)(229,231)(230,238)(232,236)(237,239)>;
G:=Group( (1,202,118,220,103,29,137,185,148,234,90,45,55,161,68)(2,69,162,56,46,91,235,149,186,138,30,104,221,119,203)(3,204,120,222,105,31,139,187,150,236,92,47,57,163,70)(4,71,164,58,48,93,237,151,188,140,32,106,223,121,205)(5,206,122,224,107,17,141,189,152,238,94,33,59,165,72)(6,73,166,60,34,95,239,153,190,142,18,108,209,123,207)(7,208,124,210,109,19,143,191,154,240,96,35,61,167,74)(8,75,168,62,36,81,225,155,192,144,20,110,211,125,193)(9,194,126,212,111,21,129,177,156,226,82,37,63,169,76)(10,77,170,64,38,83,227,157,178,130,22,112,213,127,195)(11,196,128,214,97,23,131,179,158,228,84,39,49,171,78)(12,79,172,50,40,85,229,159,180,132,24,98,215,113,197)(13,198,114,216,99,25,133,181,160,230,86,41,51,173,80)(14,65,174,52,42,87,231,145,182,134,26,100,217,115,199)(15,200,116,218,101,27,135,183,146,232,88,43,53,175,66)(16,67,176,54,44,89,233,147,184,136,28,102,219,117,201), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,86)(18,93)(19,84)(20,91)(21,82)(22,89)(23,96)(24,87)(25,94)(26,85)(27,92)(28,83)(29,90)(30,81)(31,88)(32,95)(33,198)(34,205)(35,196)(36,203)(37,194)(38,201)(39,208)(40,199)(41,206)(42,197)(43,204)(44,195)(45,202)(46,193)(47,200)(48,207)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(65,98)(66,105)(67,112)(68,103)(69,110)(70,101)(71,108)(72,99)(73,106)(74,97)(75,104)(76,111)(77,102)(78,109)(79,100)(80,107)(113,182)(114,189)(115,180)(116,187)(117,178)(118,185)(119,192)(120,183)(121,190)(122,181)(123,188)(124,179)(125,186)(126,177)(127,184)(128,191)(130,136)(131,143)(132,134)(133,141)(135,139)(138,144)(140,142)(145,172)(146,163)(147,170)(148,161)(149,168)(150,175)(151,166)(152,173)(153,164)(154,171)(155,162)(156,169)(157,176)(158,167)(159,174)(160,165)(209,223)(210,214)(211,221)(213,219)(215,217)(216,224)(218,222)(225,235)(227,233)(228,240)(229,231)(230,238)(232,236)(237,239) );
G=PermutationGroup([[(1,202,118,220,103,29,137,185,148,234,90,45,55,161,68),(2,69,162,56,46,91,235,149,186,138,30,104,221,119,203),(3,204,120,222,105,31,139,187,150,236,92,47,57,163,70),(4,71,164,58,48,93,237,151,188,140,32,106,223,121,205),(5,206,122,224,107,17,141,189,152,238,94,33,59,165,72),(6,73,166,60,34,95,239,153,190,142,18,108,209,123,207),(7,208,124,210,109,19,143,191,154,240,96,35,61,167,74),(8,75,168,62,36,81,225,155,192,144,20,110,211,125,193),(9,194,126,212,111,21,129,177,156,226,82,37,63,169,76),(10,77,170,64,38,83,227,157,178,130,22,112,213,127,195),(11,196,128,214,97,23,131,179,158,228,84,39,49,171,78),(12,79,172,50,40,85,229,159,180,132,24,98,215,113,197),(13,198,114,216,99,25,133,181,160,230,86,41,51,173,80),(14,65,174,52,42,87,231,145,182,134,26,100,217,115,199),(15,200,116,218,101,27,135,183,146,232,88,43,53,175,66),(16,67,176,54,44,89,233,147,184,136,28,102,219,117,201)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,86),(18,93),(19,84),(20,91),(21,82),(22,89),(23,96),(24,87),(25,94),(26,85),(27,92),(28,83),(29,90),(30,81),(31,88),(32,95),(33,198),(34,205),(35,196),(36,203),(37,194),(38,201),(39,208),(40,199),(41,206),(42,197),(43,204),(44,195),(45,202),(46,193),(47,200),(48,207),(49,61),(50,52),(51,59),(53,57),(54,64),(56,62),(58,60),(65,98),(66,105),(67,112),(68,103),(69,110),(70,101),(71,108),(72,99),(73,106),(74,97),(75,104),(76,111),(77,102),(78,109),(79,100),(80,107),(113,182),(114,189),(115,180),(116,187),(117,178),(118,185),(119,192),(120,183),(121,190),(122,181),(123,188),(124,179),(125,186),(126,177),(127,184),(128,191),(130,136),(131,143),(132,134),(133,141),(135,139),(138,144),(140,142),(145,172),(146,163),(147,170),(148,161),(149,168),(150,175),(151,166),(152,173),(153,164),(154,171),(155,162),(156,169),(157,176),(158,167),(159,174),(160,165),(209,223),(210,214),(211,221),(213,219),(215,217),(216,224),(218,222),(225,235),(227,233),(228,240),(229,231),(230,238),(232,236),(237,239)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 15A | 15B | 16A | 16B | 16C | 16D | 20A | 20B | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 24 | 2 | 2 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 24 | 24 | 24 | 24 | 4 | 40 | 40 | 4 | 4 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | SD32 | C5⋊D4 | D4⋊S3 | S3×D5 | D4⋊D5 | C8.6D6 | C15⋊D4 | D8.D5 | C15⋊D8 | C15⋊SD32 |
| kernel | C15⋊SD32 | C15⋊3C16 | C3×Dic20 | C5×D24 | Dic20 | C60 | D24 | C40 | C30 | C24 | C20 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C15⋊SD32 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 91 | 0 | 0 | 0 |
| 0 | 0 | 168 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 62 | 149 | 0 | 0 | 0 | 0 |
| 173 | 97 | 0 | 0 | 0 | 0 |
| 0 | 0 | 180 | 128 | 0 | 0 |
| 0 | 0 | 227 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 1 | 81 | 0 | 0 | 0 | 0 |
| 0 | 240 | 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 | 240 | 240 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,91,168,0,0,0,0,0,98,0,0,0,0,0,0,0,240,0,0,0,0,1,240],[62,173,0,0,0,0,149,97,0,0,0,0,0,0,180,227,0,0,0,0,128,61,0,0,0,0,0,0,1,240,0,0,0,0,0,240],[1,0,0,0,0,0,81,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,240,0,0,0,0,0,240] >;
C15⋊SD32 in GAP, Magma, Sage, TeX
C_{15}\rtimes {\rm SD}_{32} % in TeX
G:=Group("C15:SD32"); // GroupNames label
G:=SmallGroup(480,17);
// by ID
G=gap.SmallGroup(480,17);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,254,135,142,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^15=b^16=c^2=1,b*a*b^-1=a^-1,c*a*c=a^11,c*b*c=b^7>;
// generators/relations
Export