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G = C5⋊D48order 480 = 25·3·5

The semidirect product of C5 and D48 acting via D48/D24=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52D48, C153D16, D241D5, C30.3D8, C40.5D6, C10.6D24, C60.49D4, C20.1D12, D12010C2, C24.40D10, C120.17C22, C52C161S3, (C5×D24)⋊1C2, C31(C5⋊D16), C8.16(S3×D5), C6.1(D4⋊D5), C2.4(C5⋊D24), C4.1(C5⋊D12), C12.51(C5⋊D4), (C3×C52C16)⋊1C2, SmallGroup(480,15)

Series: Derived Chief Lower central Upper central

C1C120 — C5⋊D48
C1C5C15C30C60C120C3×C52C16 — C5⋊D48
C15C30C60C120 — C5⋊D48
C1C2C4C8

Generators and relations for C5⋊D48
 G = < a,b,c | a5=b48=c2=1, bab-1=cac=a-1, cbc=b-1 >

24C2
120C2
12C22
60C22
8S3
40S3
24D5
24C10
6D4
30D4
4D6
20D6
12D10
12C2×C10
8D15
8C5×S3
3D8
5C16
15D8
2D12
10D12
6C5×D4
6D20
4D30
4S3×C10
15D16
5C48
5D24
3C5×D8
3D40
2D60
2C5×D12
5D48
3C5⋊D16

Smallest permutation representation of C5⋊D48
On 240 points
Generators in S240
(1 182 134 57 213)(2 214 58 135 183)(3 184 136 59 215)(4 216 60 137 185)(5 186 138 61 217)(6 218 62 139 187)(7 188 140 63 219)(8 220 64 141 189)(9 190 142 65 221)(10 222 66 143 191)(11 192 144 67 223)(12 224 68 97 145)(13 146 98 69 225)(14 226 70 99 147)(15 148 100 71 227)(16 228 72 101 149)(17 150 102 73 229)(18 230 74 103 151)(19 152 104 75 231)(20 232 76 105 153)(21 154 106 77 233)(22 234 78 107 155)(23 156 108 79 235)(24 236 80 109 157)(25 158 110 81 237)(26 238 82 111 159)(27 160 112 83 239)(28 240 84 113 161)(29 162 114 85 193)(30 194 86 115 163)(31 164 116 87 195)(32 196 88 117 165)(33 166 118 89 197)(34 198 90 119 167)(35 168 120 91 199)(36 200 92 121 169)(37 170 122 93 201)(38 202 94 123 171)(39 172 124 95 203)(40 204 96 125 173)(41 174 126 49 205)(42 206 50 127 175)(43 176 128 51 207)(44 208 52 129 177)(45 178 130 53 209)(46 210 54 131 179)(47 180 132 55 211)(48 212 56 133 181)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(49 126)(50 125)(51 124)(52 123)(53 122)(54 121)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)(61 114)(62 113)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 144)(80 143)(81 142)(82 141)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(145 234)(146 233)(147 232)(148 231)(149 230)(150 229)(151 228)(152 227)(153 226)(154 225)(155 224)(156 223)(157 222)(158 221)(159 220)(160 219)(161 218)(162 217)(163 216)(164 215)(165 214)(166 213)(167 212)(168 211)(169 210)(170 209)(171 208)(172 207)(173 206)(174 205)(175 204)(176 203)(177 202)(178 201)(179 200)(180 199)(181 198)(182 197)(183 196)(184 195)(185 194)(186 193)(187 240)(188 239)(189 238)(190 237)(191 236)(192 235)

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

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

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

57 conjugacy classes

class 1 2A2B2C 3  4 5A5B 6 8A8B10A10B10C10D10E10F12A12B15A15B16A16B16C16D20A20B24A24B24C24D30A30B40A40B40C40D48A···48H60A60B60C60D120A···120H
order12223455688101010101010121215151616161620202424242430304040404048···4860606060120···120
size11241202222222222424242422441010101044222244444410···1044444···4

57 irreducible representations

dim111122222222222444444
type++++++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10D12D16C5⋊D4D24D48S3×D5D4⋊D5C5⋊D12C5⋊D16C5⋊D24C5⋊D48
kernelC5⋊D48C3×C52C16C5×D24D120C52C16C60D24C40C30C24C20C15C12C10C5C8C6C4C3C2C1
# reps111111212224448222448

Matrix representation of C5⋊D48 in GL4(𝔽241) generated by

240100
5019000
0010
0001
,
17221400
2216900
0022228
00117170
,
51100
5119000
0023954
001742
G:=sub<GL(4,GF(241))| [240,50,0,0,1,190,0,0,0,0,1,0,0,0,0,1],[172,221,0,0,214,69,0,0,0,0,222,117,0,0,28,170],[51,51,0,0,1,190,0,0,0,0,239,174,0,0,54,2] >;

C5⋊D48 in GAP, Magma, Sage, TeX

C_5\rtimes D_{48}
% in TeX

G:=Group("C5:D48");
// GroupNames label

G:=SmallGroup(480,15);
// by ID

G=gap.SmallGroup(480,15);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,85,135,142,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^5=b^48=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5⋊D48 in TeX

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