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G = D120⋊C2order 480 = 25·3·5

12nd semidirect product of D120 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.19D6, D12012C2, Dic127D5, D10.7D12, C24.46D10, C120.23C22, C60.125C23, Dic5.26D12, Dic6.22D10, D60.34C22, (C8×D5)⋊3S3, (D5×C24)⋊3C2, C53(C4○D24), C153(C4○D8), C6.10(D4×D5), C8.22(S3×D5), (C4×D5).81D6, (C6×D5).45D4, C2.15(D5×D12), C30.21(C2×D4), C52C8.34D6, C31(Q8.D10), (C5×Dic12)⋊3C2, C10.10(C2×D12), C12.28D1010C2, Dic6⋊D512C2, C20.78(C22×S3), (C3×Dic5).49D4, (D5×C12).95C22, C12.148(C22×D5), (C5×Dic6).26C22, C4.73(C2×S3×D5), (C3×C52C8).38C22, SmallGroup(480,347)

Series: Derived Chief Lower central Upper central

C1C60 — D120⋊C2
C1C5C15C30C60D5×C12C12.28D10 — D120⋊C2
C15C30C60 — D120⋊C2
C1C2C4C8

Generators and relations for D120⋊C2
 G = < a,b,c | a120=b2=c2=1, bab=a-1, cac=a49, cbc=a108b >

Subgroups: 828 in 124 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×3], C10, Dic3 [×2], C12, C12, D6 [×2], C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10, D10 [×2], C24, C24, Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C3×D5, D15 [×2], C30, C4○D8, C52C8, C40, C4×D5, C4×D5 [×2], D20 [×4], C5×Q8 [×2], C24⋊C2 [×2], D24, Dic12, C2×C24, C4○D12 [×2], C5×Dic3 [×2], C3×Dic5, C60, C6×D5, D30 [×2], C8×D5, D40, Q8⋊D5 [×2], C5×Q16, Q82D5 [×2], C4○D24, C3×C52C8, C120, D30.C2 [×2], C3⋊D20 [×2], D5×C12, C5×Dic6 [×2], D60 [×2], Q8.D10, Dic6⋊D5 [×2], D5×C24, C5×Dic12, D120, C12.28D10 [×2], D120⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C4○D8, C22×D5, C2×D12, S3×D5, D4×D5, C4○D24, C2×S3×D5, Q8.D10, D5×D12, D120⋊C2

Smallest permutation representation of D120⋊C2
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)
(1 141)(2 140)(3 139)(4 138)(5 137)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 129)(14 128)(15 127)(16 126)(17 125)(18 124)(19 123)(20 122)(21 121)(22 240)(23 239)(24 238)(25 237)(26 236)(27 235)(28 234)(29 233)(30 232)(31 231)(32 230)(33 229)(34 228)(35 227)(36 226)(37 225)(38 224)(39 223)(40 222)(41 221)(42 220)(43 219)(44 218)(45 217)(46 216)(47 215)(48 214)(49 213)(50 212)(51 211)(52 210)(53 209)(54 208)(55 207)(56 206)(57 205)(58 204)(59 203)(60 202)(61 201)(62 200)(63 199)(64 198)(65 197)(66 196)(67 195)(68 194)(69 193)(70 192)(71 191)(72 190)(73 189)(74 188)(75 187)(76 186)(77 185)(78 184)(79 183)(80 182)(81 181)(82 180)(83 179)(84 178)(85 177)(86 176)(87 175)(88 174)(89 173)(90 172)(91 171)(92 170)(93 169)(94 168)(95 167)(96 166)(97 165)(98 164)(99 163)(100 162)(101 161)(102 160)(103 159)(104 158)(105 157)(106 156)(107 155)(108 154)(109 153)(110 152)(111 151)(112 150)(113 149)(114 148)(115 147)(116 146)(117 145)(118 144)(119 143)(120 142)
(2 50)(3 99)(4 28)(5 77)(7 55)(8 104)(9 33)(10 82)(12 60)(13 109)(14 38)(15 87)(17 65)(18 114)(19 43)(20 92)(22 70)(23 119)(24 48)(25 97)(27 75)(29 53)(30 102)(32 80)(34 58)(35 107)(37 85)(39 63)(40 112)(42 90)(44 68)(45 117)(47 95)(49 73)(52 100)(54 78)(57 105)(59 83)(62 110)(64 88)(67 115)(69 93)(72 120)(74 98)(79 103)(84 108)(89 113)(94 118)(121 133)(122 182)(123 231)(124 160)(125 209)(126 138)(127 187)(128 236)(129 165)(130 214)(131 143)(132 192)(134 170)(135 219)(136 148)(137 197)(139 175)(140 224)(141 153)(142 202)(144 180)(145 229)(146 158)(147 207)(149 185)(150 234)(151 163)(152 212)(154 190)(155 239)(156 168)(157 217)(159 195)(161 173)(162 222)(164 200)(166 178)(167 227)(169 205)(171 183)(172 232)(174 210)(176 188)(177 237)(179 215)(181 193)(184 220)(186 198)(189 225)(191 203)(194 230)(196 208)(199 235)(201 213)(204 240)(206 218)(211 223)(216 228)(221 233)(226 238)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C8A8B8C8D10A10B12A12B12C12D15A15B20A20B20C20D20E20F24A24B24C24D24E24F24G24H30A30B40A40B40C40D60A60B60C60D120A···120H
order122223444445566688881010121212121515202020202020242424242424242430304040404060606060120···120
size111060602255121222210102210102222101044442424242422221010101044444444444···4

60 irreducible representations

dim1111112222222222222444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D6D10D10D12D12C4○D8C4○D24S3×D5D4×D5C2×S3×D5Q8.D10D5×D12D120⋊C2
kernelD120⋊C2Dic6⋊D5D5×C24C5×Dic12D120C12.28D10C8×D5C3×Dic5C6×D5Dic12C52C8C40C4×D5C24Dic6Dic5D10C15C5C8C6C4C3C2C1
# reps1211121112111242248222448

Matrix representation of D120⋊C2 in GL4(𝔽241) generated by

128000
2320900
0001
0024051
,
15017200
1209100
000240
002400
,
1000
17224000
0010
0051240
G:=sub<GL(4,GF(241))| [128,23,0,0,0,209,0,0,0,0,0,240,0,0,1,51],[150,120,0,0,172,91,0,0,0,0,0,240,0,0,240,0],[1,172,0,0,0,240,0,0,0,0,1,51,0,0,0,240] >;

D120⋊C2 in GAP, Magma, Sage, TeX

D_{120}\rtimes C_2
% in TeX

G:=Group("D120:C2");
// GroupNames label

G:=SmallGroup(480,347);
// by ID

G=gap.SmallGroup(480,347);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,135,142,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^120=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^49,c*b*c=a^108*b>;
// generators/relations

׿
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