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G = C24.D10order 480 = 25·3·5

9th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.7D8, C6.8D40, C154SD32, C60.26D4, C24.9D10, C40.42D6, C12.3D20, Dic201S3, D120.2C2, C120.4C22, C3⋊C163D5, C8.9(S3×D5), C33(C16⋊D5), C51(C8.6D6), C10.3(D4⋊S3), (C3×Dic20)⋊1C2, C2.6(C3⋊D40), C4.3(C3⋊D20), C20.51(C3⋊D4), (C5×C3⋊C16)⋊3C2, SmallGroup(480,19)

Series: Derived Chief Lower central Upper central

C1C120 — C24.D10
C1C5C15C30C60C120C3×Dic20 — C24.D10
C15C30C60C120 — C24.D10
C1C2C4C8

Generators and relations for C24.D10
 G = < a,b,c | a24=c2=1, b10=a15, bab-1=a17, cac=a-1, cbc=a21b9 >

120C2
20C4
60C22
40S3
24D5
10Q8
30D4
20D6
20C12
4Dic5
12D10
8D15
3C16
5Q16
15D8
10C3×Q8
10D12
2Dic10
6D20
4D30
4C3×Dic5
15SD32
5D24
5C3×Q16
3D40
3C80
2C3×Dic10
2D60
5C8.6D6
3C16⋊D5

Smallest permutation representation of C24.D10
On 240 points
Generators in S240
(1 133 199 51 103 169 21 153 219 71 123 189 41 93 239 11 143 209 61 113 179 31 83 229)(2 210 124 52 180 94 22 230 144 72 200 114 42 170 84 12 220 134 62 190 104 32 240 154)(3 135 201 53 105 171 23 155 221 73 125 191 43 95 161 13 145 211 63 115 181 33 85 231)(4 212 126 54 182 96 24 232 146 74 202 116 44 172 86 14 222 136 64 192 106 34 162 156)(5 137 203 55 107 173 25 157 223 75 127 193 45 97 163 15 147 213 65 117 183 35 87 233)(6 214 128 56 184 98 26 234 148 76 204 118 46 174 88 16 224 138 66 194 108 36 164 158)(7 139 205 57 109 175 27 159 225 77 129 195 47 99 165 17 149 215 67 119 185 37 89 235)(8 216 130 58 186 100 28 236 150 78 206 120 48 176 90 18 226 140 68 196 110 38 166 160)(9 141 207 59 111 177 29 81 227 79 131 197 49 101 167 19 151 217 69 121 187 39 91 237)(10 218 132 60 188 102 30 238 152 80 208 122 50 178 92 20 228 142 70 198 112 40 168 82)
(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 65)(2 24)(3 63)(4 22)(5 61)(6 20)(7 59)(8 18)(9 57)(10 16)(11 55)(12 14)(13 53)(15 51)(17 49)(19 47)(21 45)(23 43)(25 41)(26 80)(27 39)(28 78)(29 37)(30 76)(31 35)(32 74)(34 72)(36 70)(38 68)(40 66)(42 64)(44 62)(46 60)(48 58)(50 56)(52 54)(67 79)(69 77)(71 75)(81 185)(82 224)(83 183)(84 222)(85 181)(86 220)(87 179)(88 218)(89 177)(90 216)(91 175)(92 214)(93 173)(94 212)(95 171)(96 210)(97 169)(98 208)(99 167)(100 206)(101 165)(102 204)(103 163)(104 202)(105 161)(106 200)(107 239)(108 198)(109 237)(110 196)(111 235)(112 194)(113 233)(114 192)(115 231)(116 190)(117 229)(118 188)(119 227)(120 186)(121 225)(122 184)(123 223)(124 182)(125 221)(126 180)(127 219)(128 178)(129 217)(130 176)(131 215)(132 174)(133 213)(134 172)(135 211)(136 170)(137 209)(138 168)(139 207)(140 166)(141 205)(142 164)(143 203)(144 162)(145 201)(146 240)(147 199)(148 238)(149 197)(150 236)(151 195)(152 234)(153 193)(154 232)(155 191)(156 230)(157 189)(158 228)(159 187)(160 226)

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

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

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

66 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B12A12B12C15A15B16A16B16C16D20A20B20C20D24A24B30A30B40A···40H60A60B60C60D80A···80P120A···120H
order122344556881010121212151516161616202020202424303040···406060606080···80120···120
size111202240222222244040446666222244442···244446···64···4

66 irreducible representations

dim111122222222222444444
type++++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4SD32D20D40C16⋊D5D4⋊S3S3×D5C8.6D6C3⋊D20C3⋊D40C24.D10
kernelC24.D10C5×C3⋊C16C3×Dic20D120Dic20C60C3⋊C16C40C30C24C20C15C12C6C3C10C8C5C4C2C1
# reps1111112122244816122248

Matrix representation of C24.D10 in GL4(𝔽241) generated by

219200
6424000
0022833
00208232
,
21012900
1423100
0022016
00225127
,
1000
6424000
00190240
0019051
G:=sub<GL(4,GF(241))| [2,64,0,0,192,240,0,0,0,0,228,208,0,0,33,232],[210,142,0,0,129,31,0,0,0,0,220,225,0,0,16,127],[1,64,0,0,0,240,0,0,0,0,190,190,0,0,240,51] >;

C24.D10 in GAP, Magma, Sage, TeX

C_{24}.D_{10}
% in TeX

G:=Group("C24.D10");
// GroupNames label

G:=SmallGroup(480,19);
// by ID

G=gap.SmallGroup(480,19);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,92,590,58,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^24=c^2=1,b^10=a^15,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=a^21*b^9>;
// generators/relations

Export

Subgroup lattice of C24.D10 in TeX

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