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G = D152C16order 480 = 25·3·5

The semidirect product of D15 and C16 acting via C16/C8=C2

metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D152C16, D30.4C8, C40.50D6, C24.57D10, Dic15.4C8, C120.53C22, C53(S3×C16), C3⋊C166D5, C31(D5×C16), C157(C2×C16), C6.1(C8×D5), C52C166S3, C8.36(S3×D5), C153C8.8C4, C20.68(C4×S3), C10.10(S3×C8), C30.21(C2×C8), (C8×D15).4C2, C12.36(C4×D5), C60.137(C2×C4), (C4×D15).11C4, C2.1(D152C8), C4.16(D30.C2), (C5×C3⋊C16)⋊5C2, (C3×C52C16)⋊8C2, SmallGroup(480,9)

Series: Derived Chief Lower central Upper central

C1C15 — D152C16
C1C5C15C30C60C120C3×C52C16 — D152C16
C15 — D152C16
C1C8

Generators and relations for D152C16
 G = < a,b,c | a15=b2=c16=1, bab=a-1, cac-1=a11, cbc-1=a10b >

15C2
15C2
15C4
15C22
5S3
5S3
3D5
3D5
15C2×C4
15C8
5Dic3
5D6
3Dic5
3D10
3C16
5C16
15C2×C8
5C3⋊C8
5C4×S3
3C52C8
3C4×D5
15C2×C16
5C48
5S3×C8
3C80
3C8×D5
5S3×C16
3D5×C16

Smallest permutation representation of D152C16
On 240 points
Generators in S240
(1 59 192 151 100 20 37 207 128 92 74 228 223 137 168)(2 229 208 152 169 75 38 177 138 93 21 60 224 113 101)(3 61 178 153 102 22 39 193 114 94 76 230 209 139 170)(4 231 194 154 171 77 40 179 140 95 23 62 210 115 103)(5 63 180 155 104 24 41 195 116 96 78 232 211 141 172)(6 233 196 156 173 79 42 181 142 81 25 64 212 117 105)(7 49 182 157 106 26 43 197 118 82 80 234 213 143 174)(8 235 198 158 175 65 44 183 144 83 27 50 214 119 107)(9 51 184 159 108 28 45 199 120 84 66 236 215 129 176)(10 237 200 160 161 67 46 185 130 85 29 52 216 121 109)(11 53 186 145 110 30 47 201 122 86 68 238 217 131 162)(12 239 202 146 163 69 48 187 132 87 31 54 218 123 111)(13 55 188 147 112 32 33 203 124 88 70 240 219 133 164)(14 225 204 148 165 71 34 189 134 89 17 56 220 125 97)(15 57 190 149 98 18 35 205 126 90 72 226 221 135 166)(16 227 206 150 167 73 36 191 136 91 19 58 222 127 99)
(1 176)(2 109)(3 162)(4 111)(5 164)(6 97)(7 166)(8 99)(9 168)(10 101)(11 170)(12 103)(13 172)(14 105)(15 174)(16 107)(17 173)(18 82)(19 175)(20 84)(21 161)(22 86)(23 163)(24 88)(25 165)(26 90)(27 167)(28 92)(29 169)(30 94)(31 171)(32 96)(33 116)(34 142)(35 118)(36 144)(37 120)(38 130)(39 122)(40 132)(41 124)(42 134)(43 126)(44 136)(45 128)(46 138)(47 114)(48 140)(49 135)(50 150)(51 137)(52 152)(53 139)(54 154)(55 141)(56 156)(57 143)(58 158)(59 129)(60 160)(61 131)(62 146)(63 133)(64 148)(65 91)(66 100)(67 93)(68 102)(69 95)(70 104)(71 81)(72 106)(73 83)(74 108)(75 85)(76 110)(77 87)(78 112)(79 89)(80 98)(113 237)(115 239)(117 225)(119 227)(121 229)(123 231)(125 233)(127 235)(145 230)(147 232)(149 234)(151 236)(153 238)(155 240)(157 226)(159 228)(177 185)(178 217)(179 187)(180 219)(181 189)(182 221)(183 191)(184 223)(186 209)(188 211)(190 213)(192 215)(193 201)(194 218)(195 203)(196 220)(197 205)(198 222)(199 207)(200 224)(202 210)(204 212)(206 214)(208 216)
(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)

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

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

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

96 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B 6 8A8B8C8D8E8F8G8H10A10B12A12B15A15B16A···16H16I···16P20A20B20C20D24A24B24C24D30A30B40A···40H48A···48H60A60B60C60D80A···80P120A···120H
order1222344445568888888810101212151516···1616···162020202024242424303040···4048···486060606080···80120···120
size11151521115152221111151515152222443···35···522222222442···210···1044446···64···4

96 irreducible representations

dim11111111122222222224444
type++++++++++
imageC1C2C2C2C4C4C8C8C16S3D5D6D10C4×S3C4×D5S3×C8C8×D5S3×C16D5×C16S3×D5D30.C2D152C8D152C16
kernelD152C16C5×C3⋊C16C3×C52C16C8×D15C153C8C4×D15Dic15D30D15C52C16C3⋊C16C40C24C20C12C10C6C5C3C8C4C2C1
# reps1111224416121224488162248

Matrix representation of D152C16 in GL4(𝔽241) generated by

15400
17423900
00052
00190189
,
24018700
0100
002400
0011
,
130000
20711100
001300
000130
G:=sub<GL(4,GF(241))| [1,174,0,0,54,239,0,0,0,0,0,190,0,0,52,189],[240,0,0,0,187,1,0,0,0,0,240,1,0,0,0,1],[130,207,0,0,0,111,0,0,0,0,130,0,0,0,0,130] >;

D152C16 in GAP, Magma, Sage, TeX

D_{15}\rtimes_2C_{16}
% in TeX

G:=Group("D15:2C16");
// GroupNames label

G:=SmallGroup(480,9);
// by ID

G=gap.SmallGroup(480,9);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,36,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^15=b^2=c^16=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations

Export

Subgroup lattice of D152C16 in TeX

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