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G = C157D16order 480 = 25·3·5

1st semidirect product of C15 and D16 acting via D16/D8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C157D16, D81D15, C8.4D30, C60.3D4, D1206C2, C40.12D6, C30.39D8, C24.12D10, C120.9C22, (C5×D8)⋊1S3, (C3×D8)⋊1D5, (C15×D8)⋊1C2, C33(C5⋊D16), C53(C3⋊D16), C153C161C2, C6.17(D4⋊D5), C2.4(D4⋊D15), C4.1(C157D4), C10.17(D4⋊S3), C12.17(C5⋊D4), C20.15(C3⋊D4), SmallGroup(480,186)

Series: Derived Chief Lower central Upper central

C1C120 — C157D16
C1C5C15C30C60C120D120 — C157D16
C15C30C60C120 — C157D16
C1C2C4C8D8

Generators and relations for C157D16
 G = < a,b,c | a15=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >

8C2
120C2
4C22
60C22
8C6
40S3
8C10
24D5
2D4
30D4
4C2×C6
20D6
4C2×C10
12D10
8C30
8D15
15D8
15C16
2C3×D4
10D12
2C5×D4
6D20
4C2×C30
4D30
15D16
5D24
5C3⋊C16
3D40
3C52C16
2D4×C15
2D60
5C3⋊D16
3C5⋊D16

Smallest permutation representation of C157D16
On 240 points
Generators in S240
(1 130 111 149 69 57 48 179 206 32 226 116 221 88 170)(2 171 89 222 117 227 17 207 180 33 58 70 150 112 131)(3 132 97 151 71 59 34 181 208 18 228 118 223 90 172)(4 173 91 224 119 229 19 193 182 35 60 72 152 98 133)(5 134 99 153 73 61 36 183 194 20 230 120 209 92 174)(6 175 93 210 121 231 21 195 184 37 62 74 154 100 135)(7 136 101 155 75 63 38 185 196 22 232 122 211 94 176)(8 161 95 212 123 233 23 197 186 39 64 76 156 102 137)(9 138 103 157 77 49 40 187 198 24 234 124 213 96 162)(10 163 81 214 125 235 25 199 188 41 50 78 158 104 139)(11 140 105 159 79 51 42 189 200 26 236 126 215 82 164)(12 165 83 216 127 237 27 201 190 43 52 80 160 106 141)(13 142 107 145 65 53 44 191 202 28 238 128 217 84 166)(14 167 85 218 113 239 29 203 192 45 54 66 146 108 143)(15 144 109 147 67 55 46 177 204 30 240 114 219 86 168)(16 169 87 220 115 225 31 205 178 47 56 68 148 110 129)
(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 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(32 48)(49 234)(50 233)(51 232)(52 231)(53 230)(54 229)(55 228)(56 227)(57 226)(58 225)(59 240)(60 239)(61 238)(62 237)(63 236)(64 235)(65 120)(66 119)(67 118)(68 117)(69 116)(70 115)(71 114)(72 113)(73 128)(74 127)(75 126)(76 125)(77 124)(78 123)(79 122)(80 121)(81 102)(82 101)(83 100)(84 99)(85 98)(86 97)(87 112)(88 111)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(129 171)(130 170)(131 169)(132 168)(133 167)(134 166)(135 165)(136 164)(137 163)(138 162)(139 161)(140 176)(141 175)(142 174)(143 173)(144 172)(145 209)(146 224)(147 223)(148 222)(149 221)(150 220)(151 219)(152 218)(153 217)(154 216)(155 215)(156 214)(157 213)(158 212)(159 211)(160 210)(177 208)(178 207)(179 206)(180 205)(181 204)(182 203)(183 202)(184 201)(185 200)(186 199)(187 198)(188 197)(189 196)(190 195)(191 194)(192 193)

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

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

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

60 conjugacy classes

class 1 2A2B2C 3  4 5A5B6A6B6C8A8B10A10B10C10D10E10F 12 15A15B15C15D16A16B16C16D20A20B24A24B30A30B30C30D30E···30L40A40B40C40D60A60B60C60D120A···120H
order1222345566688101010101010121515151516161616202024243030303030···304040404060606060120···120
size1181202222288222288884222230303030444422228···8444444444···4

60 irreducible representations

dim1111222222222222444444
type+++++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4D15D16C5⋊D4D30C157D4D4⋊S3D4⋊D5C3⋊D16C5⋊D16D4⋊D15C157D16
kernelC157D16C153C16D120C15×D8C5×D8C60C3×D8C40C30C24C20D8C15C12C8C4C10C6C5C3C2C1
# reps1111112122244448122448

Matrix representation of C157D16 in GL4(𝔽241) generated by

1000
0100
0017868
0017316
,
5811900
811200
0017830
0017363
,
1000
20524000
0017830
0017363
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,178,173,0,0,68,16],[58,8,0,0,119,112,0,0,0,0,178,173,0,0,30,63],[1,205,0,0,0,240,0,0,0,0,178,173,0,0,30,63] >;

C157D16 in GAP, Magma, Sage, TeX

C_{15}\rtimes_7D_{16}
% in TeX

G:=Group("C15:7D16");
// GroupNames label

G:=SmallGroup(480,186);
// by ID

G=gap.SmallGroup(480,186);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,254,135,142,675,346,80,2693,18822]);
// Polycyclic

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

Export

Subgroup lattice of C157D16 in TeX

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