metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C30.7D8, C6.8D40, C15⋊4SD32, C60.26D4, C24.9D10, C40.42D6, C12.3D20, Dic20⋊1S3, D120.2C2, C120.4C22, C3⋊C16⋊3D5, C8.9(S3×D5), C3⋊3(C16⋊D5), C5⋊1(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
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
6D20
4D30
15SD32
5D24
3D40
3C80
2D60
Smallest permutation representation of C24.D10
►On 240 points
Generators in S240
(1 227 159 51 197 129 21 167 99 71 217 149 41 187 119 11 237 89 61 207 139 31 177 109)(2 90 218 52 140 188 22 110 238 72 160 208 42 130 178 12 100 228 62 150 198 32 120 168)(3 229 81 53 199 131 23 169 101 73 219 151 43 189 121 13 239 91 63 209 141 33 179 111)(4 92 220 54 142 190 24 112 240 74 82 210 44 132 180 14 102 230 64 152 200 34 122 170)(5 231 83 55 201 133 25 171 103 75 221 153 45 191 123 15 161 93 65 211 143 35 181 113)(6 94 222 56 144 192 26 114 162 76 84 212 46 134 182 16 104 232 66 154 202 36 124 172)(7 233 85 57 203 135 27 173 105 77 223 155 47 193 125 17 163 95 67 213 145 37 183 115)(8 96 224 58 146 194 28 116 164 78 86 214 48 136 184 18 106 234 68 156 204 38 126 174)(9 235 87 59 205 137 29 175 107 79 225 157 49 195 127 19 165 97 69 215 147 39 185 117)(10 98 226 60 148 196 30 118 166 80 88 216 50 138 186 20 108 236 70 158 206 40 128 176)
(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)(81 239)(82 198)(83 237)(84 196)(85 235)(86 194)(87 233)(88 192)(89 231)(90 190)(91 229)(92 188)(93 227)(94 186)(95 225)(96 184)(97 223)(98 182)(99 221)(100 180)(101 219)(102 178)(103 217)(104 176)(105 215)(106 174)(107 213)(108 172)(109 211)(110 170)(111 209)(112 168)(113 207)(114 166)(115 205)(116 164)(117 203)(118 162)(119 201)(120 240)(121 199)(122 238)(123 197)(124 236)(125 195)(126 234)(127 193)(128 232)(129 191)(130 230)(131 189)(132 228)(133 187)(134 226)(135 185)(136 224)(137 183)(138 222)(139 181)(140 220)(141 179)(142 218)(143 177)(144 216)(145 175)(146 214)(147 173)(148 212)(149 171)(150 210)(151 169)(152 208)(153 167)(154 206)(155 165)(156 204)(157 163)(158 202)(159 161)(160 200)
G:=sub<Sym(240)| (1,227,159,51,197,129,21,167,99,71,217,149,41,187,119,11,237,89,61,207,139,31,177,109)(2,90,218,52,140,188,22,110,238,72,160,208,42,130,178,12,100,228,62,150,198,32,120,168)(3,229,81,53,199,131,23,169,101,73,219,151,43,189,121,13,239,91,63,209,141,33,179,111)(4,92,220,54,142,190,24,112,240,74,82,210,44,132,180,14,102,230,64,152,200,34,122,170)(5,231,83,55,201,133,25,171,103,75,221,153,45,191,123,15,161,93,65,211,143,35,181,113)(6,94,222,56,144,192,26,114,162,76,84,212,46,134,182,16,104,232,66,154,202,36,124,172)(7,233,85,57,203,135,27,173,105,77,223,155,47,193,125,17,163,95,67,213,145,37,183,115)(8,96,224,58,146,194,28,116,164,78,86,214,48,136,184,18,106,234,68,156,204,38,126,174)(9,235,87,59,205,137,29,175,107,79,225,157,49,195,127,19,165,97,69,215,147,39,185,117)(10,98,226,60,148,196,30,118,166,80,88,216,50,138,186,20,108,236,70,158,206,40,128,176), (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)(81,239)(82,198)(83,237)(84,196)(85,235)(86,194)(87,233)(88,192)(89,231)(90,190)(91,229)(92,188)(93,227)(94,186)(95,225)(96,184)(97,223)(98,182)(99,221)(100,180)(101,219)(102,178)(103,217)(104,176)(105,215)(106,174)(107,213)(108,172)(109,211)(110,170)(111,209)(112,168)(113,207)(114,166)(115,205)(116,164)(117,203)(118,162)(119,201)(120,240)(121,199)(122,238)(123,197)(124,236)(125,195)(126,234)(127,193)(128,232)(129,191)(130,230)(131,189)(132,228)(133,187)(134,226)(135,185)(136,224)(137,183)(138,222)(139,181)(140,220)(141,179)(142,218)(143,177)(144,216)(145,175)(146,214)(147,173)(148,212)(149,171)(150,210)(151,169)(152,208)(153,167)(154,206)(155,165)(156,204)(157,163)(158,202)(159,161)(160,200)>;
G:=Group( (1,227,159,51,197,129,21,167,99,71,217,149,41,187,119,11,237,89,61,207,139,31,177,109)(2,90,218,52,140,188,22,110,238,72,160,208,42,130,178,12,100,228,62,150,198,32,120,168)(3,229,81,53,199,131,23,169,101,73,219,151,43,189,121,13,239,91,63,209,141,33,179,111)(4,92,220,54,142,190,24,112,240,74,82,210,44,132,180,14,102,230,64,152,200,34,122,170)(5,231,83,55,201,133,25,171,103,75,221,153,45,191,123,15,161,93,65,211,143,35,181,113)(6,94,222,56,144,192,26,114,162,76,84,212,46,134,182,16,104,232,66,154,202,36,124,172)(7,233,85,57,203,135,27,173,105,77,223,155,47,193,125,17,163,95,67,213,145,37,183,115)(8,96,224,58,146,194,28,116,164,78,86,214,48,136,184,18,106,234,68,156,204,38,126,174)(9,235,87,59,205,137,29,175,107,79,225,157,49,195,127,19,165,97,69,215,147,39,185,117)(10,98,226,60,148,196,30,118,166,80,88,216,50,138,186,20,108,236,70,158,206,40,128,176), (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)(81,239)(82,198)(83,237)(84,196)(85,235)(86,194)(87,233)(88,192)(89,231)(90,190)(91,229)(92,188)(93,227)(94,186)(95,225)(96,184)(97,223)(98,182)(99,221)(100,180)(101,219)(102,178)(103,217)(104,176)(105,215)(106,174)(107,213)(108,172)(109,211)(110,170)(111,209)(112,168)(113,207)(114,166)(115,205)(116,164)(117,203)(118,162)(119,201)(120,240)(121,199)(122,238)(123,197)(124,236)(125,195)(126,234)(127,193)(128,232)(129,191)(130,230)(131,189)(132,228)(133,187)(134,226)(135,185)(136,224)(137,183)(138,222)(139,181)(140,220)(141,179)(142,218)(143,177)(144,216)(145,175)(146,214)(147,173)(148,212)(149,171)(150,210)(151,169)(152,208)(153,167)(154,206)(155,165)(156,204)(157,163)(158,202)(159,161)(160,200) );
G=PermutationGroup([[(1,227,159,51,197,129,21,167,99,71,217,149,41,187,119,11,237,89,61,207,139,31,177,109),(2,90,218,52,140,188,22,110,238,72,160,208,42,130,178,12,100,228,62,150,198,32,120,168),(3,229,81,53,199,131,23,169,101,73,219,151,43,189,121,13,239,91,63,209,141,33,179,111),(4,92,220,54,142,190,24,112,240,74,82,210,44,132,180,14,102,230,64,152,200,34,122,170),(5,231,83,55,201,133,25,171,103,75,221,153,45,191,123,15,161,93,65,211,143,35,181,113),(6,94,222,56,144,192,26,114,162,76,84,212,46,134,182,16,104,232,66,154,202,36,124,172),(7,233,85,57,203,135,27,173,105,77,223,155,47,193,125,17,163,95,67,213,145,37,183,115),(8,96,224,58,146,194,28,116,164,78,86,214,48,136,184,18,106,234,68,156,204,38,126,174),(9,235,87,59,205,137,29,175,107,79,225,157,49,195,127,19,165,97,69,215,147,39,185,117),(10,98,226,60,148,196,30,118,166,80,88,216,50,138,186,20,108,236,70,158,206,40,128,176)], [(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),(81,239),(82,198),(83,237),(84,196),(85,235),(86,194),(87,233),(88,192),(89,231),(90,190),(91,229),(92,188),(93,227),(94,186),(95,225),(96,184),(97,223),(98,182),(99,221),(100,180),(101,219),(102,178),(103,217),(104,176),(105,215),(106,174),(107,213),(108,172),(109,211),(110,170),(111,209),(112,168),(113,207),(114,166),(115,205),(116,164),(117,203),(118,162),(119,201),(120,240),(121,199),(122,238),(123,197),(124,236),(125,195),(126,234),(127,193),(128,232),(129,191),(130,230),(131,189),(132,228),(133,187),(134,226),(135,185),(136,224),(137,183),(138,222),(139,181),(140,220),(141,179),(142,218),(143,177),(144,216),(145,175),(146,214),(147,173),(148,212),(149,171),(150,210),(151,169),(152,208),(153,167),(154,206),(155,165),(156,204),(157,163),(158,202),(159,161),(160,200)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D | 80A | ··· | 80P | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| size | 1 | 1 | 120 | 2 | 2 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 40 | 40 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | SD32 | D20 | D40 | C16⋊D5 | D4⋊S3 | S3×D5 | C8.6D6 | C3⋊D20 | C3⋊D40 | C24.D10 |
| kernel | C24.D10 | C5×C3⋊C16 | C3×Dic20 | D120 | Dic20 | C60 | C3⋊C16 | C40 | C30 | C24 | C20 | C15 | C12 | C6 | C3 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 1 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C24.D10 ►in GL4(𝔽241) generated by
| 2 | 192 | 0 | 0 |
| 64 | 240 | 0 | 0 |
| 0 | 0 | 228 | 33 |
| 0 | 0 | 208 | 232 |
| 210 | 129 | 0 | 0 |
| 142 | 31 | 0 | 0 |
| 0 | 0 | 220 | 16 |
| 0 | 0 | 225 | 127 |
| 1 | 0 | 0 | 0 |
| 64 | 240 | 0 | 0 |
| 0 | 0 | 190 | 240 |
| 0 | 0 | 190 | 51 |
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