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G = C15⋊D16order 480 = 25·3·5

1st semidirect product of C15 and D16 acting via D16/C8=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C151D16, D403S3, D243D5, C40.4D6, C30.1D8, C24.4D10, C60.72D4, C120.32C22, (C3×D40)⋊6C2, (C5×D24)⋊5C2, C52(C3⋊D16), C32(C5⋊D16), C153C164C2, C8.25(S3×D5), C6.8(D4⋊D5), C10.8(D4⋊S3), C12.1(C5⋊D4), C4.1(C15⋊D4), C2.4(C15⋊D8), C20.1(C3⋊D4), SmallGroup(480,13)

Series: Derived Chief Lower central Upper central

C1C120 — C15⋊D16
C1C5C15C30C60C120C3×D40 — C15⋊D16
C15C30C60C120 — C15⋊D16
C1C2C4C8

Generators and relations for C15⋊D16
 G = < a,b,c | a15=b16=c2=1, bab-1=a-1, cac=a4, cbc=b-1 >

24C2
40C2
12C22
20C22
8S3
40C6
8D5
24C10
6D4
10D4
4D6
20C2×C6
4D10
12C2×C10
8C5×S3
8C3×D5
3D8
5D8
15C16
2D12
10C3×D4
2D20
6C5×D4
4S3×C10
4C6×D5
15D16
5C3×D8
5C3⋊C16
3C5×D8
3C52C16
2C5×D12
2C3×D20
5C3⋊D16
3C5⋊D16

Smallest permutation representation of C15⋊D16
On 240 points
Generators in S240
(1 156 142 22 57 115 101 65 216 228 192 167 200 86 33)(2 34 87 201 168 177 229 217 66 102 116 58 23 143 157)(3 158 144 24 59 117 103 67 218 230 178 169 202 88 35)(4 36 89 203 170 179 231 219 68 104 118 60 25 129 159)(5 160 130 26 61 119 105 69 220 232 180 171 204 90 37)(6 38 91 205 172 181 233 221 70 106 120 62 27 131 145)(7 146 132 28 63 121 107 71 222 234 182 173 206 92 39)(8 40 93 207 174 183 235 223 72 108 122 64 29 133 147)(9 148 134 30 49 123 109 73 224 236 184 175 208 94 41)(10 42 95 193 176 185 237 209 74 110 124 50 31 135 149)(11 150 136 32 51 125 111 75 210 238 186 161 194 96 43)(12 44 81 195 162 187 239 211 76 112 126 52 17 137 151)(13 152 138 18 53 127 97 77 212 240 188 163 196 82 45)(14 46 83 197 164 189 225 213 78 98 128 54 19 139 153)(15 154 140 20 55 113 99 79 214 226 190 165 198 84 47)(16 48 85 199 166 191 227 215 80 100 114 56 21 141 155)
(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 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 205)(18 204)(19 203)(20 202)(21 201)(22 200)(23 199)(24 198)(25 197)(26 196)(27 195)(28 194)(29 193)(30 208)(31 207)(32 206)(33 167)(34 166)(35 165)(36 164)(37 163)(38 162)(39 161)(40 176)(41 175)(42 174)(43 173)(44 172)(45 171)(46 170)(47 169)(48 168)(49 148)(50 147)(51 146)(52 145)(53 160)(54 159)(55 158)(56 157)(57 156)(58 155)(59 154)(60 153)(61 152)(62 151)(63 150)(64 149)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(97 232)(98 231)(99 230)(100 229)(101 228)(102 227)(103 226)(104 225)(105 240)(106 239)(107 238)(108 237)(109 236)(110 235)(111 234)(112 233)(113 117)(114 116)(118 128)(119 127)(120 126)(121 125)(122 124)(129 213)(130 212)(131 211)(132 210)(133 209)(134 224)(135 223)(136 222)(137 221)(138 220)(139 219)(140 218)(141 217)(142 216)(143 215)(144 214)(177 191)(178 190)(179 189)(180 188)(181 187)(182 186)(183 185)

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

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

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

48 conjugacy classes

class 1 2A2B2C 3  4 5A5B6A6B6C8A8B10A10B10C10D10E10F 12 15A15B16A16B16C16D20A20B24A24B30A30B40A40B40C40D60A60B60C60D120A···120H
order1222345566688101010101010121515161616162020242430304040404060606060120···120
size11244022222404022222424242444430303030444444444444444···4

48 irreducible representations

dim111122222222244444444
type+++++++++++++++-+
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4D16C5⋊D4D4⋊S3S3×D5D4⋊D5C3⋊D16C15⋊D4C5⋊D16C15⋊D8C15⋊D16
kernelC15⋊D16C153C16C3×D40C5×D24D40C60D24C40C30C24C20C15C12C10C8C6C5C4C3C2C1
# reps111111212224412222448

Matrix representation of C15⋊D16 in GL6(𝔽241)

01890000
51510000
001516200
00022500
000010
000001
,
1721450000
145690000
002014000
002074000
0000172204
00002815
,
51520000
1911900000
0012600
00024000
0000994
0000194232

G:=sub<GL(6,GF(241))| [0,51,0,0,0,0,189,51,0,0,0,0,0,0,15,0,0,0,0,0,162,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[172,145,0,0,0,0,145,69,0,0,0,0,0,0,201,207,0,0,0,0,40,40,0,0,0,0,0,0,172,28,0,0,0,0,204,15],[51,191,0,0,0,0,52,190,0,0,0,0,0,0,1,0,0,0,0,0,26,240,0,0,0,0,0,0,9,194,0,0,0,0,94,232] >;

C15⋊D16 in GAP, Magma, Sage, TeX

C_{15}\rtimes D_{16}
% in TeX

G:=Group("C15:D16");
// GroupNames label

G:=SmallGroup(480,13);
// by ID

G=gap.SmallGroup(480,13);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,254,135,142,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^15=b^16=c^2=1,b*a*b^-1=a^-1,c*a*c=a^4,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C15⋊D16 in TeX

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