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G = C48⋊D5order 480 = 25·3·5

2nd semidirect product of C48 and D5 acting via D5/C5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C802S3, C482D5, C2402C2, C162D15, C6.2D40, C4.2D60, C157SD32, C2.4D120, C10.2D24, C30.23D8, C40.66D6, C8.14D30, Dic601C2, D120.1C2, C24.66D10, C60.158D4, C20.27D12, C12.27D20, C120.79C22, C31(C16⋊D5), C51(C48⋊C2), SmallGroup(480,160)

Series: Derived Chief Lower central Upper central

C1C120 — C48⋊D5
C1C5C15C30C60C120D120 — C48⋊D5
C15C30C60C120 — C48⋊D5
C1C2C4C8C16

Generators and relations for C48⋊D5
 G = < a,b,c | a48=b5=c2=1, ab=ba, cac=a23, cbc=b-1 >

120C2
60C22
60C4
40S3
24D5
30D4
30Q8
20D6
20Dic3
12Dic5
12D10
8D15
15Q16
15D8
10D12
10Dic6
6D20
6Dic10
4Dic15
4D30
15SD32
5Dic12
5D24
3Dic20
3D40
2D60
2Dic30
5C48⋊C2
3C16⋊D5

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

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

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

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

123 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B12A12B15A15B15C15D16A16B16C16D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H48A···48H60A···60H80A···80P120A···120P240A···240AF
order1223445568810101212151515151616161620202020242424243030303040···4048···4860···6080···80120···120240···240
size1112022120222222222222222222222222222222···22···22···22···22···22···2

123 irreducible representations

dim1111222222222222222222
type++++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10D12D15SD32D20D24D30D40C48⋊C2D60C16⋊D5D120C48⋊D5
kernelC48⋊D5C240D120Dic60C80C60C48C40C30C24C20C16C15C12C10C8C6C5C4C3C2C1
# reps1111112122244444888161632

Matrix representation of C48⋊D5 in GL4(𝔽241) generated by

3411600
12515000
00113163
007872
,
1000
0100
0001
00240189
,
0100
1000
0001
0010
G:=sub<GL(4,GF(241))| [34,125,0,0,116,150,0,0,0,0,113,78,0,0,163,72],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C48⋊D5 in GAP, Magma, Sage, TeX

C_{48}\rtimes D_5
% in TeX

G:=Group("C48:D5");
// GroupNames label

G:=SmallGroup(480,160);
// by ID

G=gap.SmallGroup(480,160);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,590,58,675,80,2693,18822]);
// Polycyclic

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

Export

Subgroup lattice of C48⋊D5 in TeX

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