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G = C5×C48⋊C2order 480 = 25·3·5

Direct product of C5 and C48⋊C2

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C48⋊C2, C806S3, C2408C2, C482C10, C159SD32, C40.76D6, C30.33D8, D24.1C10, C10.14D24, C20.40D12, C60.178D4, Dic121C10, C120.94C22, C162(C5×S3), C6.2(C5×D8), C31(C5×SD32), C2.4(C5×D24), C4.2(C5×D12), C8.14(S3×C10), (C5×D24).3C2, C12.25(C5×D4), C24.15(C2×C10), (C5×Dic12)⋊9C2, SmallGroup(480,119)

Series: Derived Chief Lower central Upper central

C1C24 — C5×C48⋊C2
C1C3C6C12C24C120C5×D24 — C5×C48⋊C2
C3C6C12C24 — C5×C48⋊C2
C1C10C20C40C80

Generators and relations for C5×C48⋊C2
 G = < a,b,c | a5=b48=c2=1, ab=ba, ac=ca, cbc=b23 >

24C2
12C22
12C4
8S3
24C10
6D4
6Q8
4Dic3
4D6
12C2×C10
12C20
8C5×S3
3Q16
3D8
2Dic6
2D12
6C5×D4
6C5×Q8
4C5×Dic3
4S3×C10
3SD32
3C5×D8
3C5×Q16
2C5×D12
2C5×Dic6
3C5×SD32

Smallest permutation representation of C5×C48⋊C2
On 240 points
Generators in S240
(1 84 206 181 143)(2 85 207 182 144)(3 86 208 183 97)(4 87 209 184 98)(5 88 210 185 99)(6 89 211 186 100)(7 90 212 187 101)(8 91 213 188 102)(9 92 214 189 103)(10 93 215 190 104)(11 94 216 191 105)(12 95 217 192 106)(13 96 218 145 107)(14 49 219 146 108)(15 50 220 147 109)(16 51 221 148 110)(17 52 222 149 111)(18 53 223 150 112)(19 54 224 151 113)(20 55 225 152 114)(21 56 226 153 115)(22 57 227 154 116)(23 58 228 155 117)(24 59 229 156 118)(25 60 230 157 119)(26 61 231 158 120)(27 62 232 159 121)(28 63 233 160 122)(29 64 234 161 123)(30 65 235 162 124)(31 66 236 163 125)(32 67 237 164 126)(33 68 238 165 127)(34 69 239 166 128)(35 70 240 167 129)(36 71 193 168 130)(37 72 194 169 131)(38 73 195 170 132)(39 74 196 171 133)(40 75 197 172 134)(41 76 198 173 135)(42 77 199 174 136)(43 78 200 175 137)(44 79 201 176 138)(45 80 202 177 139)(46 81 203 178 140)(47 82 204 179 141)(48 83 205 180 142)
(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)
(2 24)(3 47)(4 22)(5 45)(6 20)(7 43)(8 18)(9 41)(10 16)(11 39)(12 14)(13 37)(15 35)(17 33)(19 31)(21 29)(23 27)(26 48)(28 46)(30 44)(32 42)(34 40)(36 38)(49 95)(50 70)(51 93)(52 68)(53 91)(54 66)(55 89)(56 64)(57 87)(58 62)(59 85)(61 83)(63 81)(65 79)(67 77)(69 75)(71 73)(72 96)(74 94)(76 92)(78 90)(80 88)(82 86)(97 141)(98 116)(99 139)(100 114)(101 137)(102 112)(103 135)(104 110)(105 133)(106 108)(107 131)(109 129)(111 127)(113 125)(115 123)(117 121)(118 144)(120 142)(122 140)(124 138)(126 136)(128 134)(130 132)(145 169)(146 192)(147 167)(148 190)(149 165)(150 188)(151 163)(152 186)(153 161)(154 184)(155 159)(156 182)(158 180)(160 178)(162 176)(164 174)(166 172)(168 170)(171 191)(173 189)(175 187)(177 185)(179 183)(193 195)(194 218)(196 216)(197 239)(198 214)(199 237)(200 212)(201 235)(202 210)(203 233)(204 208)(205 231)(207 229)(209 227)(211 225)(213 223)(215 221)(217 219)(220 240)(222 238)(224 236)(226 234)(228 232)

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

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

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

135 conjugacy classes

class 1 2A2B 3 4A4B5A5B5C5D 6 8A8B10A10B10C10D10E10F10G10H12A12B15A15B15C15D16A16B16C16D20A20B20C20D20E20F20G20H24A24B24C24D30A30B30C30D40A···40H48A···48H60A···60H80A···80P120A···120P240A···240AF
order12234455556881010101010101010121215151515161616162020202020202020242424243030303040···4048···4860···6080···80120···120240···240
size1124222411112221111242424242222222222222224242424222222222···22···22···22···22···22···2

135 irreducible representations

dim111111112222222222222222
type++++++++++
imageC1C2C2C2C5C10C10C10S3D4D6D8D12C5×S3SD32C5×D4D24S3×C10C5×D8C48⋊C2C5×D12C5×SD32C5×D24C5×C48⋊C2
kernelC5×C48⋊C2C240C5×D24C5×Dic12C48⋊C2C48D24Dic12C80C60C40C30C20C16C15C12C10C8C6C5C4C3C2C1
# reps111144441112244444888161632

Matrix representation of C5×C48⋊C2 in GL2(𝔽241) generated by

910
091
,
76100
141176
,
01
10
G:=sub<GL(2,GF(241))| [91,0,0,91],[76,141,100,176],[0,1,1,0] >;

C5×C48⋊C2 in GAP, Magma, Sage, TeX

C_5\times C_{48}\rtimes C_2
% in TeX

G:=Group("C5xC48:C2");
// GroupNames label

G:=SmallGroup(480,119);
// by ID

G=gap.SmallGroup(480,119);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,309,428,3923,80,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c|a^5=b^48=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^23>;
// generators/relations

Export

Subgroup lattice of C5×C48⋊C2 in TeX

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