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## G = C15⋊SD32order 480 = 25·3·5

### 2nd semidirect product of C15 and SD32 acting via SD32/C8=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C120 — C15⋊SD32
 Chief series C1 — C5 — C15 — C30 — C60 — C120 — C3×Dic20 — C15⋊SD32
 Lower central C15 — C30 — C60 — C120 — C15⋊SD32
 Upper central C1 — C2 — C4 — C8

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

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

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

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

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

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