Copied to
clipboard

G = C153C16order 240 = 24·3·5

1st semidirect product of C15 and C16 acting via C16/C8=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C153C16, C30.3C8, C60.8C4, C40.2S3, C8.2D15, C24.3D5, C120.3C2, C4.2Dic15, C20.5Dic3, C12.2Dic5, C3⋊(C52C16), C52(C3⋊C16), C6.(C52C8), C2.(C153C8), C10.2(C3⋊C8), SmallGroup(240,3)

Series: Derived Chief Lower central Upper central

C1C15 — C153C16
C1C5C15C30C60C120 — C153C16
C15 — C153C16
C1C8

Generators and relations for C153C16
 G = < a,b | a15=b16=1, bab-1=a-1 >

15C16
5C3⋊C16
3C52C16

Smallest permutation representation of C153C16
Regular action on 240 points
Generators in S240
(1 91 46 164 142 150 201 100 115 218 55 72 235 29 188)(2 189 30 236 73 56 219 116 101 202 151 143 165 47 92)(3 93 48 166 144 152 203 102 117 220 57 74 237 31 190)(4 191 32 238 75 58 221 118 103 204 153 129 167 33 94)(5 95 34 168 130 154 205 104 119 222 59 76 239 17 192)(6 177 18 240 77 60 223 120 105 206 155 131 169 35 96)(7 81 36 170 132 156 207 106 121 224 61 78 225 19 178)(8 179 20 226 79 62 209 122 107 208 157 133 171 37 82)(9 83 38 172 134 158 193 108 123 210 63 80 227 21 180)(10 181 22 228 65 64 211 124 109 194 159 135 173 39 84)(11 85 40 174 136 160 195 110 125 212 49 66 229 23 182)(12 183 24 230 67 50 213 126 111 196 145 137 175 41 86)(13 87 42 176 138 146 197 112 127 214 51 68 231 25 184)(14 185 26 232 69 52 215 128 97 198 147 139 161 43 88)(15 89 44 162 140 148 199 98 113 216 53 70 233 27 186)(16 187 28 234 71 54 217 114 99 200 149 141 163 45 90)
(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)| (1,91,46,164,142,150,201,100,115,218,55,72,235,29,188)(2,189,30,236,73,56,219,116,101,202,151,143,165,47,92)(3,93,48,166,144,152,203,102,117,220,57,74,237,31,190)(4,191,32,238,75,58,221,118,103,204,153,129,167,33,94)(5,95,34,168,130,154,205,104,119,222,59,76,239,17,192)(6,177,18,240,77,60,223,120,105,206,155,131,169,35,96)(7,81,36,170,132,156,207,106,121,224,61,78,225,19,178)(8,179,20,226,79,62,209,122,107,208,157,133,171,37,82)(9,83,38,172,134,158,193,108,123,210,63,80,227,21,180)(10,181,22,228,65,64,211,124,109,194,159,135,173,39,84)(11,85,40,174,136,160,195,110,125,212,49,66,229,23,182)(12,183,24,230,67,50,213,126,111,196,145,137,175,41,86)(13,87,42,176,138,146,197,112,127,214,51,68,231,25,184)(14,185,26,232,69,52,215,128,97,198,147,139,161,43,88)(15,89,44,162,140,148,199,98,113,216,53,70,233,27,186)(16,187,28,234,71,54,217,114,99,200,149,141,163,45,90), (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( (1,91,46,164,142,150,201,100,115,218,55,72,235,29,188)(2,189,30,236,73,56,219,116,101,202,151,143,165,47,92)(3,93,48,166,144,152,203,102,117,220,57,74,237,31,190)(4,191,32,238,75,58,221,118,103,204,153,129,167,33,94)(5,95,34,168,130,154,205,104,119,222,59,76,239,17,192)(6,177,18,240,77,60,223,120,105,206,155,131,169,35,96)(7,81,36,170,132,156,207,106,121,224,61,78,225,19,178)(8,179,20,226,79,62,209,122,107,208,157,133,171,37,82)(9,83,38,172,134,158,193,108,123,210,63,80,227,21,180)(10,181,22,228,65,64,211,124,109,194,159,135,173,39,84)(11,85,40,174,136,160,195,110,125,212,49,66,229,23,182)(12,183,24,230,67,50,213,126,111,196,145,137,175,41,86)(13,87,42,176,138,146,197,112,127,214,51,68,231,25,184)(14,185,26,232,69,52,215,128,97,198,147,139,161,43,88)(15,89,44,162,140,148,199,98,113,216,53,70,233,27,186)(16,187,28,234,71,54,217,114,99,200,149,141,163,45,90), (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([(1,91,46,164,142,150,201,100,115,218,55,72,235,29,188),(2,189,30,236,73,56,219,116,101,202,151,143,165,47,92),(3,93,48,166,144,152,203,102,117,220,57,74,237,31,190),(4,191,32,238,75,58,221,118,103,204,153,129,167,33,94),(5,95,34,168,130,154,205,104,119,222,59,76,239,17,192),(6,177,18,240,77,60,223,120,105,206,155,131,169,35,96),(7,81,36,170,132,156,207,106,121,224,61,78,225,19,178),(8,179,20,226,79,62,209,122,107,208,157,133,171,37,82),(9,83,38,172,134,158,193,108,123,210,63,80,227,21,180),(10,181,22,228,65,64,211,124,109,194,159,135,173,39,84),(11,85,40,174,136,160,195,110,125,212,49,66,229,23,182),(12,183,24,230,67,50,213,126,111,196,145,137,175,41,86),(13,87,42,176,138,146,197,112,127,214,51,68,231,25,184),(14,185,26,232,69,52,215,128,97,198,147,139,161,43,88),(15,89,44,162,140,148,199,98,113,216,53,70,233,27,186),(16,187,28,234,71,54,217,114,99,200,149,141,163,45,90)], [(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)])

C153C16 is a maximal subgroup of
D5×C3⋊C16  S3×C52C16  C40.51D6  C40.52D6  C15⋊D16  C40.D6  C15⋊SD32  C15⋊Q32  C16×D15  C80⋊S3  C60.7C8  C157D16  D8.D15  C8.6D30  C157Q32
C153C16 is a maximal quotient of
C153C32

72 conjugacy classes

class 1  2  3 4A4B5A5B 6 8A8B8C8D10A10B12A12B15A15B15C15D16A···16H20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order123445568888101012121515151516···1620202020242424243030303040···4060···60120···120
size1121122211112222222215···152222222222222···22···22···2

72 irreducible representations

dim11111222222222222
type++++--+-
imageC1C2C4C8C16S3D5Dic3Dic5C3⋊C8D15C52C8C3⋊C16Dic15C52C16C153C8C153C16
kernelC153C16C120C60C30C15C40C24C20C12C10C8C6C5C4C3C2C1
# reps112481212244448816

Matrix representation of C153C16 in GL4(𝔽241) generated by

24018900
525200
00154
00174239
,
4512600
19619600
00210165
001931
G:=sub<GL(4,GF(241))| [240,52,0,0,189,52,0,0,0,0,1,174,0,0,54,239],[45,196,0,0,126,196,0,0,0,0,210,19,0,0,165,31] >;

C153C16 in GAP, Magma, Sage, TeX

C_{15}\rtimes_3C_{16}
% in TeX

G:=Group("C15:3C16");
// GroupNames label

G:=SmallGroup(240,3);
// by ID

G=gap.SmallGroup(240,3);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,12,31,50,964,6917]);
// Polycyclic

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

Export

Subgroup lattice of C153C16 in TeX

׿
×
𝔽