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G = C5⋊Dic12order 240 = 24·3·5

The semidirect product of C5 and Dic12 acting via Dic12/Dic6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C153Q16, C20.8D6, C52Dic12, C30.12D4, C10.9D12, C12.25D10, Dic6.1D5, C60.18C22, Dic30.4C2, C52C8.1S3, C4.11(S3×D5), C31(C5⋊Q16), C6.4(C5⋊D4), C2.7(C5⋊D12), (C5×Dic6).1C2, (C3×C52C8).1C2, SmallGroup(240,24)

Series: Derived Chief Lower central Upper central

C1C60 — C5⋊Dic12
C1C5C15C30C60C3×C52C8 — C5⋊Dic12
C15C30C60 — C5⋊Dic12
C1C2C4

Generators and relations for C5⋊Dic12
 G = < a,b,c | a5=b24=1, c2=b12, bab-1=a-1, ac=ca, cbc-1=b-1 >

6C4
30C4
3Q8
5C8
15Q8
2Dic3
10Dic3
6Dic5
6C20
15Q16
5C24
5Dic6
3Dic10
3C5×Q8
2C5×Dic3
2Dic15
5Dic12
3C5⋊Q16

Smallest permutation representation of C5⋊Dic12
Regular action on 240 points
Generators in S240
(1 202 69 172 227)(2 228 173 70 203)(3 204 71 174 229)(4 230 175 72 205)(5 206 49 176 231)(6 232 177 50 207)(7 208 51 178 233)(8 234 179 52 209)(9 210 53 180 235)(10 236 181 54 211)(11 212 55 182 237)(12 238 183 56 213)(13 214 57 184 239)(14 240 185 58 215)(15 216 59 186 217)(16 218 187 60 193)(17 194 61 188 219)(18 220 189 62 195)(19 196 63 190 221)(20 222 191 64 197)(21 198 65 192 223)(22 224 169 66 199)(23 200 67 170 225)(24 226 171 68 201)(25 128 165 74 107)(26 108 75 166 129)(27 130 167 76 109)(28 110 77 168 131)(29 132 145 78 111)(30 112 79 146 133)(31 134 147 80 113)(32 114 81 148 135)(33 136 149 82 115)(34 116 83 150 137)(35 138 151 84 117)(36 118 85 152 139)(37 140 153 86 119)(38 120 87 154 141)(39 142 155 88 97)(40 98 89 156 143)(41 144 157 90 99)(42 100 91 158 121)(43 122 159 92 101)(44 102 93 160 123)(45 124 161 94 103)(46 104 95 162 125)(47 126 163 96 105)(48 106 73 164 127)
(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)
(1 161 13 149)(2 160 14 148)(3 159 15 147)(4 158 16 146)(5 157 17 145)(6 156 18 168)(7 155 19 167)(8 154 20 166)(9 153 21 165)(10 152 22 164)(11 151 23 163)(12 150 24 162)(25 180 37 192)(26 179 38 191)(27 178 39 190)(28 177 40 189)(29 176 41 188)(30 175 42 187)(31 174 43 186)(32 173 44 185)(33 172 45 184)(34 171 46 183)(35 170 47 182)(36 169 48 181)(49 99 61 111)(50 98 62 110)(51 97 63 109)(52 120 64 108)(53 119 65 107)(54 118 66 106)(55 117 67 105)(56 116 68 104)(57 115 69 103)(58 114 70 102)(59 113 71 101)(60 112 72 100)(73 211 85 199)(74 210 86 198)(75 209 87 197)(76 208 88 196)(77 207 89 195)(78 206 90 194)(79 205 91 193)(80 204 92 216)(81 203 93 215)(82 202 94 214)(83 201 95 213)(84 200 96 212)(121 218 133 230)(122 217 134 229)(123 240 135 228)(124 239 136 227)(125 238 137 226)(126 237 138 225)(127 236 139 224)(128 235 140 223)(129 234 141 222)(130 233 142 221)(131 232 143 220)(132 231 144 219)

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

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

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

C5⋊Dic12 is a maximal subgroup of
D5×Dic12  Dic60⋊C2  C24.2D10  C40.31D6  C20.60D12  C20.D12  D12.33D10  C60.10C23  D30.9D4  D20.24D6  D20.10D6  S3×C5⋊Q16  D15⋊Q16  D20.28D6  D20.17D6
C5⋊Dic12 is a maximal quotient of
C10.Dic12  Dic3015C4  C60.8Q8

33 conjugacy classes

class 1  2  3 4A4B4C5A5B 6 8A8B10A10B12A12B15A15B20A20B20C20D20E20F24A24B24C24D30A30B60A60B60C60D
order1234445568810101212151520202020202024242424303060606060
size112212602221010222244441212121210101010444444

33 irreducible representations

dim11112222222224444
type++++++++-++-+-+-
imageC1C2C2C2S3D4D5D6Q16D10D12C5⋊D4Dic12S3×D5C5⋊Q16C5⋊D12C5⋊Dic12
kernelC5⋊Dic12C3×C52C8C5×Dic6Dic30C52C8C30Dic6C20C15C12C10C6C5C4C3C2C1
# reps11111121222442224

Matrix representation of C5⋊Dic12 in GL4(𝔽241) generated by

1000
0100
002401
0050190
,
911400
12713600
0022842
0023713
,
1407000
17110100
002400
000240
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[9,127,0,0,114,136,0,0,0,0,228,237,0,0,42,13],[140,171,0,0,70,101,0,0,0,0,240,0,0,0,0,240] >;

C5⋊Dic12 in GAP, Magma, Sage, TeX

C_5\rtimes {\rm Dic}_{12}
% in TeX

G:=Group("C5:Dic12");
// GroupNames label

G:=SmallGroup(240,24);
// by ID

G=gap.SmallGroup(240,24);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,73,55,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^5=b^24=1,c^2=b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5⋊Dic12 in TeX

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