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G = D8×C30order 480 = 25·3·5

Direct product of C30 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: D8×C30, C60.198D4, C12046C22, C60.292C23, (C2×C8)⋊3C30, C82(C2×C30), (C2×C40)⋊11C6, C4012(C2×C6), (C2×C24)⋊8C10, D41(C2×C30), (C2×D4)⋊4C30, C4.6(D4×C15), (C2×C120)⋊24C2, C2410(C2×C10), (D4×C10)⋊13C6, (D4×C30)⋊31C2, (C6×D4)⋊13C10, C2.11(D4×C30), C10.74(C6×D4), C20.41(C3×D4), C6.74(D4×C10), C12.41(C5×D4), (C2×C30).195D4, C30.457(C2×D4), C4.1(C22×C30), (D4×C15)⋊41C22, C20.44(C22×C6), C22.14(D4×C15), (C2×C60).582C22, C12.44(C22×C10), (C5×D4)⋊10(C2×C6), (C2×C6).52(C5×D4), (C3×D4)⋊10(C2×C10), (C2×C4).26(C2×C30), (C2×C10).52(C3×D4), (C2×C20).128(C2×C6), (C2×C12).129(C2×C10), SmallGroup(480,937)

Series: Derived Chief Lower central Upper central

C1C4 — D8×C30
C1C2C4C20C60D4×C15C15×D8 — D8×C30
C1C2C4 — D8×C30
C1C2×C30C2×C60 — D8×C30

Generators and relations for D8×C30
 G = < a,b,c | a30=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 280 in 152 conjugacy classes, 88 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, C6, C6 [×2], C6 [×4], C8 [×2], C2×C4, D4 [×4], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×4], C12 [×2], C2×C6, C2×C6 [×8], C15, C2×C8, D8 [×4], C2×D4 [×2], C20 [×2], C2×C10, C2×C10 [×8], C24 [×2], C2×C12, C3×D4 [×4], C3×D4 [×2], C22×C6 [×2], C30, C30 [×2], C30 [×4], C2×D8, C40 [×2], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×C10 [×2], C2×C24, C3×D8 [×4], C6×D4 [×2], C60 [×2], C2×C30, C2×C30 [×8], C2×C40, C5×D8 [×4], D4×C10 [×2], C6×D8, C120 [×2], C2×C60, D4×C15 [×4], D4×C15 [×2], C22×C30 [×2], C10×D8, C2×C120, C15×D8 [×4], D4×C30 [×2], D8×C30
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], C23, C10 [×7], C2×C6 [×7], C15, D8 [×2], C2×D4, C2×C10 [×7], C3×D4 [×2], C22×C6, C30 [×7], C2×D8, C5×D4 [×2], C22×C10, C3×D8 [×2], C6×D4, C2×C30 [×7], C5×D8 [×2], D4×C10, C6×D8, D4×C15 [×2], C22×C30, C10×D8, C15×D8 [×2], D4×C30, D8×C30

Smallest permutation representation of D8×C30
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)
(1 158 111 90 232 188 45 139)(2 159 112 61 233 189 46 140)(3 160 113 62 234 190 47 141)(4 161 114 63 235 191 48 142)(5 162 115 64 236 192 49 143)(6 163 116 65 237 193 50 144)(7 164 117 66 238 194 51 145)(8 165 118 67 239 195 52 146)(9 166 119 68 240 196 53 147)(10 167 120 69 211 197 54 148)(11 168 91 70 212 198 55 149)(12 169 92 71 213 199 56 150)(13 170 93 72 214 200 57 121)(14 171 94 73 215 201 58 122)(15 172 95 74 216 202 59 123)(16 173 96 75 217 203 60 124)(17 174 97 76 218 204 31 125)(18 175 98 77 219 205 32 126)(19 176 99 78 220 206 33 127)(20 177 100 79 221 207 34 128)(21 178 101 80 222 208 35 129)(22 179 102 81 223 209 36 130)(23 180 103 82 224 210 37 131)(24 151 104 83 225 181 38 132)(25 152 105 84 226 182 39 133)(26 153 106 85 227 183 40 134)(27 154 107 86 228 184 41 135)(28 155 108 87 229 185 42 136)(29 156 109 88 230 186 43 137)(30 157 110 89 231 187 44 138)
(1 217)(2 218)(3 219)(4 220)(5 221)(6 222)(7 223)(8 224)(9 225)(10 226)(11 227)(12 228)(13 229)(14 230)(15 231)(16 232)(17 233)(18 234)(19 235)(20 236)(21 237)(22 238)(23 239)(24 240)(25 211)(26 212)(27 213)(28 214)(29 215)(30 216)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 174)(62 175)(63 176)(64 177)(65 178)(66 179)(67 180)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 159)(77 160)(78 161)(79 162)(80 163)(81 164)(82 165)(83 166)(84 167)(85 168)(86 169)(87 170)(88 171)(89 172)(90 173)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(121 185)(122 186)(123 187)(124 188)(125 189)(126 190)(127 191)(128 192)(129 193)(130 194)(131 195)(132 196)(133 197)(134 198)(135 199)(136 200)(137 201)(138 202)(139 203)(140 204)(141 205)(142 206)(143 207)(144 208)(145 209)(146 210)(147 181)(148 182)(149 183)(150 184)

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

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

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

210 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B5A5B5C5D6A···6F6G···6N8A8B8C8D10A···10L10M···10AB12A12B12C12D15A···15H20A···20H24A···24H30A···30X30Y···30BD40A···40P60A···60P120A···120AF
order12222222334455556···66···6888810···1010···101212121215···1520···2024···2430···3030···3040···4060···60120···120
size11114444112211111···14···422221···14···422221···12···22···21···14···42···22···22···2

210 irreducible representations

dim1111111111111111222222222222
type+++++++
imageC1C2C2C2C3C5C6C6C6C10C10C10C15C30C30C30D4D4D8C3×D4C3×D4C5×D4C5×D4C3×D8C5×D8D4×C15D4×C15C15×D8
kernelD8×C30C2×C120C15×D8D4×C30C10×D8C6×D8C2×C40C5×D8D4×C10C2×C24C3×D8C6×D4C2×D8C2×C8D8C2×D4C60C2×C30C30C20C2×C10C12C2×C6C10C6C4C22C2
# reps114224284416888321611422448168832

Matrix representation of D8×C30 in GL3(𝔽241) generated by

1600
02050
00205
,
100
02211
02190
,
100
0240240
001
G:=sub<GL(3,GF(241))| [16,0,0,0,205,0,0,0,205],[1,0,0,0,22,219,0,11,0],[1,0,0,0,240,0,0,240,1] >;

D8×C30 in GAP, Magma, Sage, TeX

D_8\times C_{30}
% in TeX

G:=Group("D8xC30");
// GroupNames label

G:=SmallGroup(480,937);
// by ID

G=gap.SmallGroup(480,937);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,15125,7572,124]);
// Polycyclic

G:=Group<a,b,c|a^30=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽