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G = C157Q16order 240 = 24·3·5

1st semidirect product of C15 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C157Q16, C4.4D30, C30.37D4, C20.12D6, Q8.2D15, C12.12D10, C60.4C22, Dic30.2C2, C53(C3⋊Q16), C33(C5⋊Q16), (C5×Q8).3S3, (C3×Q8).1D5, C153C8.1C2, (Q8×C15).1C2, C6.19(C5⋊D4), C2.7(C157D4), C10.19(C3⋊D4), SmallGroup(240,79)

Series: Derived Chief Lower central Upper central

C1C60 — C157Q16
C1C5C15C30C60Dic30 — C157Q16
C15C30C60 — C157Q16
C1C2C4Q8

Generators and relations for C157Q16
 G = < a,b,c | a15=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

2C4
30C4
15Q8
15C8
2C12
10Dic3
2C20
6Dic5
15Q16
5Dic6
5C3⋊C8
3Dic10
3C52C8
2C60
2Dic15
5C3⋊Q16
3C5⋊Q16

Smallest permutation representation of C157Q16
Regular action on 240 points
Generators in S240
(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)
(1 218 35 198 16 237 50 182)(2 217 36 197 17 236 51 181)(3 216 37 196 18 235 52 195)(4 215 38 210 19 234 53 194)(5 214 39 209 20 233 54 193)(6 213 40 208 21 232 55 192)(7 212 41 207 22 231 56 191)(8 211 42 206 23 230 57 190)(9 225 43 205 24 229 58 189)(10 224 44 204 25 228 59 188)(11 223 45 203 26 227 60 187)(12 222 31 202 27 226 46 186)(13 221 32 201 28 240 47 185)(14 220 33 200 29 239 48 184)(15 219 34 199 30 238 49 183)(61 162 106 130 76 175 91 136)(62 161 107 129 77 174 92 150)(63 160 108 128 78 173 93 149)(64 159 109 127 79 172 94 148)(65 158 110 126 80 171 95 147)(66 157 111 125 81 170 96 146)(67 156 112 124 82 169 97 145)(68 155 113 123 83 168 98 144)(69 154 114 122 84 167 99 143)(70 153 115 121 85 166 100 142)(71 152 116 135 86 180 101 141)(72 151 117 134 87 179 102 140)(73 165 118 133 88 178 103 139)(74 164 119 132 89 177 104 138)(75 163 120 131 90 176 105 137)
(1 80 16 65)(2 81 17 66)(3 82 18 67)(4 83 19 68)(5 84 20 69)(6 85 21 70)(7 86 22 71)(8 87 23 72)(9 88 24 73)(10 89 25 74)(11 90 26 75)(12 76 27 61)(13 77 28 62)(14 78 29 63)(15 79 30 64)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)(121 232 142 213)(122 233 143 214)(123 234 144 215)(124 235 145 216)(125 236 146 217)(126 237 147 218)(127 238 148 219)(128 239 149 220)(129 240 150 221)(130 226 136 222)(131 227 137 223)(132 228 138 224)(133 229 139 225)(134 230 140 211)(135 231 141 212)(151 190 179 206)(152 191 180 207)(153 192 166 208)(154 193 167 209)(155 194 168 210)(156 195 169 196)(157 181 170 197)(158 182 171 198)(159 183 172 199)(160 184 173 200)(161 185 174 201)(162 186 175 202)(163 187 176 203)(164 188 177 204)(165 189 178 205)

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

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

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

C157Q16 is a maximal subgroup of
D5×C3⋊Q16  D20.13D6  S3×C5⋊Q16  Dic10.26D6  D12.27D10  D20.14D6  D20.27D6  D20.28D6  SD16⋊D15  D4.5D30  Q16×D15  Q16⋊D15  Q8.11D30  D4.8D30  D4.9D30
C157Q16 is a maximal quotient of
C60.1Q8  Dic309C4  Q82Dic15

42 conjugacy classes

class 1  2  3 4A4B4C5A5B 6 8A8B10A10B12A12B12C15A15B15C15D20A···20F30A30B30C30D60A···60L
order1234445568810101212121515151520···203030303060···60
size112246022230302244422224···422224···4

42 irreducible representations

dim111122222222222444
type++++++++-+++---
imageC1C2C2C2S3D4D5D6Q16D10C3⋊D4D15C5⋊D4D30C157D4C3⋊Q16C5⋊Q16C157Q16
kernelC157Q16C153C8Dic30Q8×C15C5×Q8C30C3×Q8C20C15C12C10Q8C6C4C2C5C3C1
# reps111111212224448124

Matrix representation of C157Q16 in GL6(𝔽241)

2251440000
0150000
001895100
00189000
000010
000001
,
122350000
431190000
0020410800
001483700
000023011
0000230230
,
100000
010000
001000
000100
000056128
0000128185

G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,144,15,0,0,0,0,0,0,189,189,0,0,0,0,51,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[122,43,0,0,0,0,35,119,0,0,0,0,0,0,204,148,0,0,0,0,108,37,0,0,0,0,0,0,230,230,0,0,0,0,11,230],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,56,128,0,0,0,0,128,185] >;

C157Q16 in GAP, Magma, Sage, TeX

C_{15}\rtimes_7Q_{16}
% in TeX

G:=Group("C15:7Q16");
// GroupNames label

G:=SmallGroup(240,79);
// by ID

G=gap.SmallGroup(240,79);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,55,218,116,50,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^15=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C157Q16 in TeX

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