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## G = C10×C24⋊C2order 480 = 25·3·5

### Direct product of C10 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C10×C24⋊C2
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C10×D12 — C10×C24⋊C2
 Lower central C3 — C6 — C12 — C10×C24⋊C2
 Upper central C1 — C2×C10 — C2×C20 — C2×C40

Generators and relations for C10×C24⋊C2
G = < a,b,c | a10=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Subgroups: 388 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C24, Dic6, Dic6, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C24⋊C2, C2×C24, C2×Dic6, C2×D12, C5×Dic3, C60, S3×C10, C2×C30, C2×C40, C5×SD16, D4×C10, Q8×C10, C2×C24⋊C2, C120, C5×Dic6, C5×Dic6, C5×D12, C5×D12, C10×Dic3, C2×C60, S3×C2×C10, C10×SD16, C5×C24⋊C2, C2×C120, C10×Dic6, C10×D12, C10×C24⋊C2
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, SD16, C2×D4, C2×C10, D12, C22×S3, C5×S3, C2×SD16, C5×D4, C22×C10, C24⋊C2, C2×D12, S3×C10, C5×SD16, D4×C10, C2×C24⋊C2, C5×D12, S3×C2×C10, C10×SD16, C5×C24⋊C2, C10×D12, C10×C24⋊C2

Smallest permutation representation of C10×C24⋊C2
On 240 points
Generators in S240
(1 83 98 213 55 222 168 43 137 190)(2 84 99 214 56 223 145 44 138 191)(3 85 100 215 57 224 146 45 139 192)(4 86 101 216 58 225 147 46 140 169)(5 87 102 193 59 226 148 47 141 170)(6 88 103 194 60 227 149 48 142 171)(7 89 104 195 61 228 150 25 143 172)(8 90 105 196 62 229 151 26 144 173)(9 91 106 197 63 230 152 27 121 174)(10 92 107 198 64 231 153 28 122 175)(11 93 108 199 65 232 154 29 123 176)(12 94 109 200 66 233 155 30 124 177)(13 95 110 201 67 234 156 31 125 178)(14 96 111 202 68 235 157 32 126 179)(15 73 112 203 69 236 158 33 127 180)(16 74 113 204 70 237 159 34 128 181)(17 75 114 205 71 238 160 35 129 182)(18 76 115 206 72 239 161 36 130 183)(19 77 116 207 49 240 162 37 131 184)(20 78 117 208 50 217 163 38 132 185)(21 79 118 209 51 218 164 39 133 186)(22 80 119 210 52 219 165 40 134 187)(23 81 120 211 53 220 166 41 135 188)(24 82 97 212 54 221 167 42 136 189)
(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)
(1 222)(2 233)(3 220)(4 231)(5 218)(6 229)(7 240)(8 227)(9 238)(10 225)(11 236)(12 223)(13 234)(14 221)(15 232)(16 219)(17 230)(18 217)(19 228)(20 239)(21 226)(22 237)(23 224)(24 235)(25 116)(26 103)(27 114)(28 101)(29 112)(30 99)(31 110)(32 97)(33 108)(34 119)(35 106)(36 117)(37 104)(38 115)(39 102)(40 113)(41 100)(42 111)(43 98)(44 109)(45 120)(46 107)(47 118)(48 105)(49 172)(50 183)(51 170)(52 181)(53 192)(54 179)(55 190)(56 177)(57 188)(58 175)(59 186)(60 173)(61 184)(62 171)(63 182)(64 169)(65 180)(66 191)(67 178)(68 189)(69 176)(70 187)(71 174)(72 185)(73 154)(74 165)(75 152)(76 163)(77 150)(78 161)(79 148)(80 159)(81 146)(82 157)(83 168)(84 155)(85 166)(86 153)(87 164)(88 151)(89 162)(90 149)(91 160)(92 147)(93 158)(94 145)(95 156)(96 167)(121 205)(122 216)(123 203)(124 214)(125 201)(126 212)(127 199)(128 210)(129 197)(130 208)(131 195)(132 206)(133 193)(134 204)(135 215)(136 202)(137 213)(138 200)(139 211)(140 198)(141 209)(142 196)(143 207)(144 194)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 2 3 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 12 12 2 2 2 12 12 1 1 1 1 2 2 2 2 2 2 2 1 ··· 1 12 ··· 12 2 2 2 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 D4 D4 D6 D6 SD16 D12 D12 C5×S3 C5×D4 C5×D4 C24⋊C2 S3×C10 S3×C10 C5×SD16 C5×D12 C5×D12 C5×C24⋊C2 kernel C10×C24⋊C2 C5×C24⋊C2 C2×C120 C10×Dic6 C10×D12 C2×C24⋊C2 C24⋊C2 C2×C24 C2×Dic6 C2×D12 C2×C40 C60 C2×C30 C40 C2×C20 C30 C20 C2×C10 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 4 16 4 4 4 1 1 1 2 1 4 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C10×C24⋊C2 in GL3(𝔽241) generated by

 240 0 0 0 36 0 0 0 36
,
 240 0 0 0 213 175 0 66 147
,
 1 0 0 0 240 1 0 0 1
G:=sub<GL(3,GF(241))| [240,0,0,0,36,0,0,0,36],[240,0,0,0,213,66,0,175,147],[1,0,0,0,240,0,0,1,1] >;

C10×C24⋊C2 in GAP, Magma, Sage, TeX

C_{10}\times C_{24}\rtimes C_2
% in TeX

G:=Group("C10xC24:C2");
// GroupNames label

G:=SmallGroup(480,781);
// by ID

G=gap.SmallGroup(480,781);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,226,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c|a^10=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations

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