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G = D5×C48order 480 = 25·3·5

Direct product of C48 and D5

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

Aliases: D5×C48, C804C6, C24010C2, D10.4C24, C24.80D10, Dic5.4C24, C120.98C22, C53(C2×C48), C52C166C6, C1511(C2×C16), (C6×D5).8C8, C6.16(C8×D5), C2.1(D5×C24), C8.19(C6×D5), C52C8.7C12, C30.48(C2×C8), C40.19(C2×C6), (C8×D5).11C6, (C4×D5).9C12, C4.16(D5×C12), C12.86(C4×D5), C10.10(C2×C24), C20.42(C2×C12), C60.208(C2×C4), (D5×C12).19C4, (D5×C24).22C2, (C3×Dic5).8C8, (C3×C52C16)⋊13C2, (C3×C52C8).14C4, SmallGroup(480,75)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C48
C1C5C10C20C40C120D5×C24 — D5×C48
C5 — D5×C48
C1C48

Generators and relations for D5×C48
 G = < a,b,c | a48=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C4
5C22
5C6
5C6
5C8
5C2×C4
5C12
5C2×C6
5C16
5C2×C8
5C24
5C2×C12
5C2×C16
5C48
5C2×C24
5C2×C48

Smallest permutation representation of D5×C48
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)
(1 190 128 196 91)(2 191 129 197 92)(3 192 130 198 93)(4 145 131 199 94)(5 146 132 200 95)(6 147 133 201 96)(7 148 134 202 49)(8 149 135 203 50)(9 150 136 204 51)(10 151 137 205 52)(11 152 138 206 53)(12 153 139 207 54)(13 154 140 208 55)(14 155 141 209 56)(15 156 142 210 57)(16 157 143 211 58)(17 158 144 212 59)(18 159 97 213 60)(19 160 98 214 61)(20 161 99 215 62)(21 162 100 216 63)(22 163 101 217 64)(23 164 102 218 65)(24 165 103 219 66)(25 166 104 220 67)(26 167 105 221 68)(27 168 106 222 69)(28 169 107 223 70)(29 170 108 224 71)(30 171 109 225 72)(31 172 110 226 73)(32 173 111 227 74)(33 174 112 228 75)(34 175 113 229 76)(35 176 114 230 77)(36 177 115 231 78)(37 178 116 232 79)(38 179 117 233 80)(39 180 118 234 81)(40 181 119 235 82)(41 182 120 236 83)(42 183 121 237 84)(43 184 122 238 85)(44 185 123 239 86)(45 186 124 240 87)(46 187 125 193 88)(47 188 126 194 89)(48 189 127 195 90)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(97 121)(98 122)(99 123)(100 124)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)(145 223)(146 224)(147 225)(148 226)(149 227)(150 228)(151 229)(152 230)(153 231)(154 232)(155 233)(156 234)(157 235)(158 236)(159 237)(160 238)(161 239)(162 240)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 199)(170 200)(171 201)(172 202)(173 203)(174 204)(175 205)(176 206)(177 207)(178 208)(179 209)(180 210)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)

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

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

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

192 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D5A5B6A6B6C6D6E6F8A8B8C8D8E8F8G8H10A10B12A12B12C12D12E12F12G12H15A15B15C15D16A···16H16I···16P20A20B20C20D24A···24H24I···24P30A30B30C30D40A···40H48A···48P48Q···48AF60A···60H80A···80P120A···120P240A···240AF
order12223344445566666688888888101012121212121212121515151516···1616···162020202024···2424···243030303040···4048···4848···4860···6080···80120···120240···240
size11551111552211555511115555221111555522221···15···522221···15···522222···21···15···52···22···22···22···2

192 irreducible representations

dim1111111111111111112222222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C8C8C12C12C16C24C24C48D5D10C3×D5C4×D5C6×D5C8×D5D5×C12D5×C16D5×C24D5×C48
kernelD5×C48C3×C52C16C240D5×C24D5×C16C3×C52C8D5×C12C52C16C80C8×D5C3×Dic5C6×D5C52C8C4×D5C3×D5Dic5D10D5C48C24C16C12C8C6C4C3C2C1
# reps111122222244441688322244488161632

Matrix representation of D5×C48 in GL3(𝔽241) generated by

19700
0320
0032
,
100
001
0240189
,
100
001
010
G:=sub<GL(3,GF(241))| [197,0,0,0,32,0,0,0,32],[1,0,0,0,0,240,0,1,189],[1,0,0,0,0,1,0,1,0] >;

D5×C48 in GAP, Magma, Sage, TeX

D_5\times C_{48}
% in TeX

G:=Group("D5xC48");
// GroupNames label

G:=SmallGroup(480,75);
// by ID

G=gap.SmallGroup(480,75);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,92,80,102,18822]);
// Polycyclic

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

Export

Subgroup lattice of D5×C48 in TeX

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