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G = D1205C2order 480 = 25·3·5

5th semidirect product of D120 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1205C2, D6.3D20, C40.46D6, Dic207S3, C24.19D10, C120.8C22, C60.100C23, Dic3.14D20, Dic10.22D6, D60.29C22, (S3×C8)⋊3D5, (S3×C40)⋊3C2, C156(C4○D8), C3⋊C8.30D10, C8.13(S3×D5), C2.15(S3×D20), C30.25(C2×D4), C6.10(C2×D20), C10.10(S3×D4), C33(D407C2), C51(D24⋊C2), (C3×Dic20)⋊3C2, (C4×S3).41D10, (S3×C10).22D4, D60⋊C210C2, C15⋊SD1612C2, (C5×Dic3).25D4, C12.78(C22×D5), (S3×C20).47C22, C20.150(C22×S3), (C3×Dic10).26C22, C4.99(C2×S3×D5), (C5×C3⋊C8).34C22, SmallGroup(480,351)

Series: Derived Chief Lower central Upper central

C1C60 — D1205C2
C1C5C15C30C60C3×Dic10D60⋊C2 — D1205C2
C15C30C60 — D1205C2
C1C2C4C8

Generators and relations for D1205C2
 G = < a,b,c | a120=b2=c2=1, bab=a-1, cac=a41, cbc=a100b >

Subgroups: 828 in 124 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×3], C6, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, Dic3, C12, C12 [×2], D6, D6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20, C20, D10 [×2], C2×C10, C3⋊C8, C24, C4×S3, C4×S3 [×2], D12 [×4], C3×Q8 [×2], C5×S3, D15 [×2], C30, C4○D8, C40, C40, Dic10 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, S3×C8, D24, Q82S3 [×2], C3×Q16, Q83S3 [×2], C5×Dic3, C3×Dic5 [×2], C60, S3×C10, D30 [×2], C40⋊C2 [×2], D40, Dic20, C2×C40, C4○D20 [×2], D24⋊C2, C5×C3⋊C8, C120, D30.C2 [×2], C5⋊D12 [×2], C3×Dic10 [×2], S3×C20, D60 [×2], D407C2, C15⋊SD16 [×2], C3×Dic20, S3×C40, D120, D60⋊C2 [×2], D1205C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, D20 [×2], C22×D5, S3×D4, S3×D5, C2×D20, D24⋊C2, C2×S3×D5, D407C2, S3×D20, D1205C2

Smallest permutation representation of D1205C2
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)
(1 185)(2 184)(3 183)(4 182)(5 181)(6 180)(7 179)(8 178)(9 177)(10 176)(11 175)(12 174)(13 173)(14 172)(15 171)(16 170)(17 169)(18 168)(19 167)(20 166)(21 165)(22 164)(23 163)(24 162)(25 161)(26 160)(27 159)(28 158)(29 157)(30 156)(31 155)(32 154)(33 153)(34 152)(35 151)(36 150)(37 149)(38 148)(39 147)(40 146)(41 145)(42 144)(43 143)(44 142)(45 141)(46 140)(47 139)(48 138)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 125)(62 124)(63 123)(64 122)(65 121)(66 240)(67 239)(68 238)(69 237)(70 236)(71 235)(72 234)(73 233)(74 232)(75 231)(76 230)(77 229)(78 228)(79 227)(80 226)(81 225)(82 224)(83 223)(84 222)(85 221)(86 220)(87 219)(88 218)(89 217)(90 216)(91 215)(92 214)(93 213)(94 212)(95 211)(96 210)(97 209)(98 208)(99 207)(100 206)(101 205)(102 204)(103 203)(104 202)(105 201)(106 200)(107 199)(108 198)(109 197)(110 196)(111 195)(112 194)(113 193)(114 192)(115 191)(116 190)(117 189)(118 188)(119 187)(120 186)
(2 42)(3 83)(5 45)(6 86)(8 48)(9 89)(11 51)(12 92)(14 54)(15 95)(17 57)(18 98)(20 60)(21 101)(23 63)(24 104)(26 66)(27 107)(29 69)(30 110)(32 72)(33 113)(35 75)(36 116)(38 78)(39 119)(41 81)(44 84)(47 87)(50 90)(53 93)(56 96)(59 99)(62 102)(65 105)(68 108)(71 111)(74 114)(77 117)(80 120)(121 221)(122 142)(123 183)(124 224)(125 145)(126 186)(127 227)(128 148)(129 189)(130 230)(131 151)(132 192)(133 233)(134 154)(135 195)(136 236)(137 157)(138 198)(139 239)(140 160)(141 201)(143 163)(144 204)(146 166)(147 207)(149 169)(150 210)(152 172)(153 213)(155 175)(156 216)(158 178)(159 219)(161 181)(162 222)(164 184)(165 225)(167 187)(168 228)(170 190)(171 231)(173 193)(174 234)(176 196)(177 237)(179 199)(180 240)(182 202)(185 205)(188 208)(191 211)(194 214)(197 217)(200 220)(203 223)(206 226)(209 229)(212 232)(215 235)(218 238)

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

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

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

69 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B 6 8A8B8C8D10A10B10C10D10E10F12A12B12C15A15B20A20B20C20D20E20F20G20H24A24B30A30B40A···40H40I···40P60A60B60C60D120A···120H
order122223444445568888101010101010121212151520202020202020202424303040···4040···4060606060120···120
size116606022332020222226622666644040442222666644442···26···644444···4

69 irreducible representations

dim1111112222222222222444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D10C4○D8D20D20D407C2S3×D4S3×D5D24⋊C2C2×S3×D5S3×D20D1205C2
kernelD1205C2C15⋊SD16C3×Dic20S3×C40D120D60⋊C2Dic20C5×Dic3S3×C10S3×C8C40Dic10C3⋊C8C24C4×S3C15Dic3D6C3C10C8C5C4C2C1
# reps12111211121222244416122248

Matrix representation of D1205C2 in GL4(𝔽241) generated by

200000
04700
00239151
002331
,
012500
27000
002400
002331
,
1000
024000
00240151
0001
G:=sub<GL(4,GF(241))| [200,0,0,0,0,47,0,0,0,0,239,233,0,0,151,1],[0,27,0,0,125,0,0,0,0,0,240,233,0,0,0,1],[1,0,0,0,0,240,0,0,0,0,240,0,0,0,151,1] >;

D1205C2 in GAP, Magma, Sage, TeX

D_{120}\rtimes_5C_2
% in TeX

G:=Group("D120:5C2");
// GroupNames label

G:=SmallGroup(480,351);
// by ID

G=gap.SmallGroup(480,351);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,142,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^120=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^41,c*b*c=a^100*b>;
// generators/relations

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