direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×C5⋊2C8, C20.40C23, C23.4Dic5, (C2×C10)⋊5C8, C10⋊4(C2×C8), C5⋊4(C22×C8), C20.63(C2×C4), (C2×C20).23C4, (C2×C4).99D10, (C2×C4).9Dic5, (C22×C4).9D5, C4.14(C2×Dic5), C4.40(C22×D5), (C22×C10).10C4, (C22×C20).12C2, C10.33(C22×C4), C2.1(C22×Dic5), (C2×C20).112C22, C22.11(C2×Dic5), (C2×C10).51(C2×C4), SmallGroup(160,141)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C22×C5⋊2C8 |
Generators and relations for C22×C5⋊2C8
G = < a,b,c,d | a2=b2=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 120 in 76 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, C22×C4, C20, C20, C2×C10, C22×C8, C5⋊2C8, C2×C20, C22×C10, C2×C5⋊2C8, C22×C20, C22×C5⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, Dic5, D10, C22×C8, C5⋊2C8, C2×Dic5, C22×D5, C2×C5⋊2C8, C22×Dic5, C22×C5⋊2C8
►Regular action on 160 points
Generators in S160
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 105)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(33 76)(34 77)(35 78)(36 79)(37 80)(38 73)(39 74)(40 75)(41 142)(42 143)(43 144)(44 137)(45 138)(46 139)(47 140)(48 141)(49 150)(50 151)(51 152)(52 145)(53 146)(54 147)(55 148)(56 149)(57 158)(58 159)(59 160)(60 153)(61 154)(62 155)(63 156)(64 157)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(81 126)(82 127)(83 128)(84 121)(85 122)(86 123)(87 124)(88 125)(89 134)(90 135)(91 136)(92 129)(93 130)(94 131)(95 132)(96 133)
(1 144)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 80)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 110)(34 111)(35 112)(36 105)(37 106)(38 107)(39 108)(40 109)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 121)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 129)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 145)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 153)(97 120)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)
(1 11 65 147 155)(2 156 148 66 12)(3 13 67 149 157)(4 158 150 68 14)(5 15 69 151 159)(6 160 152 70 16)(7 9 71 145 153)(8 154 146 72 10)(17 86 94 140 78)(18 79 141 95 87)(19 88 96 142 80)(20 73 143 89 81)(21 82 90 144 74)(22 75 137 91 83)(23 84 92 138 76)(24 77 139 93 85)(25 112 117 50 58)(26 59 51 118 105)(27 106 119 52 60)(28 61 53 120 107)(29 108 113 54 62)(30 63 55 114 109)(31 110 115 56 64)(32 57 49 116 111)(33 100 121 129 45)(34 46 130 122 101)(35 102 123 131 47)(36 48 132 124 103)(37 104 125 133 41)(38 42 134 126 97)(39 98 127 135 43)(40 44 136 128 99)
(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)
G:=sub<Sym(160)| (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,105)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(33,76)(34,77)(35,78)(36,79)(37,80)(38,73)(39,74)(40,75)(41,142)(42,143)(43,144)(44,137)(45,138)(46,139)(47,140)(48,141)(49,150)(50,151)(51,152)(52,145)(53,146)(54,147)(55,148)(56,149)(57,158)(58,159)(59,160)(60,153)(61,154)(62,155)(63,156)(64,157)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(81,126)(82,127)(83,128)(84,121)(85,122)(86,123)(87,124)(88,125)(89,134)(90,135)(91,136)(92,129)(93,130)(94,131)(95,132)(96,133), (1,144)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,80)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,110)(34,111)(35,112)(36,105)(37,106)(38,107)(39,108)(40,109)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,121)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,129)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,145)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,153)(97,120)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119), (1,11,65,147,155)(2,156,148,66,12)(3,13,67,149,157)(4,158,150,68,14)(5,15,69,151,159)(6,160,152,70,16)(7,9,71,145,153)(8,154,146,72,10)(17,86,94,140,78)(18,79,141,95,87)(19,88,96,142,80)(20,73,143,89,81)(21,82,90,144,74)(22,75,137,91,83)(23,84,92,138,76)(24,77,139,93,85)(25,112,117,50,58)(26,59,51,118,105)(27,106,119,52,60)(28,61,53,120,107)(29,108,113,54,62)(30,63,55,114,109)(31,110,115,56,64)(32,57,49,116,111)(33,100,121,129,45)(34,46,130,122,101)(35,102,123,131,47)(36,48,132,124,103)(37,104,125,133,41)(38,42,134,126,97)(39,98,127,135,43)(40,44,136,128,99), (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)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,105)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(33,76)(34,77)(35,78)(36,79)(37,80)(38,73)(39,74)(40,75)(41,142)(42,143)(43,144)(44,137)(45,138)(46,139)(47,140)(48,141)(49,150)(50,151)(51,152)(52,145)(53,146)(54,147)(55,148)(56,149)(57,158)(58,159)(59,160)(60,153)(61,154)(62,155)(63,156)(64,157)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(81,126)(82,127)(83,128)(84,121)(85,122)(86,123)(87,124)(88,125)(89,134)(90,135)(91,136)(92,129)(93,130)(94,131)(95,132)(96,133), (1,144)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,80)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,69)(18,70)(19,71)(20,72)(21,65)(22,66)(23,67)(24,68)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,110)(34,111)(35,112)(36,105)(37,106)(38,107)(39,108)(40,109)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,121)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,129)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,145)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,153)(97,120)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119), (1,11,65,147,155)(2,156,148,66,12)(3,13,67,149,157)(4,158,150,68,14)(5,15,69,151,159)(6,160,152,70,16)(7,9,71,145,153)(8,154,146,72,10)(17,86,94,140,78)(18,79,141,95,87)(19,88,96,142,80)(20,73,143,89,81)(21,82,90,144,74)(22,75,137,91,83)(23,84,92,138,76)(24,77,139,93,85)(25,112,117,50,58)(26,59,51,118,105)(27,106,119,52,60)(28,61,53,120,107)(29,108,113,54,62)(30,63,55,114,109)(31,110,115,56,64)(32,57,49,116,111)(33,100,121,129,45)(34,46,130,122,101)(35,102,123,131,47)(36,48,132,124,103)(37,104,125,133,41)(38,42,134,126,97)(39,98,127,135,43)(40,44,136,128,99), (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) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,105),(17,102),(18,103),(19,104),(20,97),(21,98),(22,99),(23,100),(24,101),(33,76),(34,77),(35,78),(36,79),(37,80),(38,73),(39,74),(40,75),(41,142),(42,143),(43,144),(44,137),(45,138),(46,139),(47,140),(48,141),(49,150),(50,151),(51,152),(52,145),(53,146),(54,147),(55,148),(56,149),(57,158),(58,159),(59,160),(60,153),(61,154),(62,155),(63,156),(64,157),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(81,126),(82,127),(83,128),(84,121),(85,122),(86,123),(87,124),(88,125),(89,134),(90,135),(91,136),(92,129),(93,130),(94,131),(95,132),(96,133)], [(1,144),(2,137),(3,138),(4,139),(5,140),(6,141),(7,142),(8,143),(9,80),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,69),(18,70),(19,71),(20,72),(21,65),(22,66),(23,67),(24,68),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,110),(34,111),(35,112),(36,105),(37,106),(38,107),(39,108),(40,109),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,121),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,129),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,145),(89,154),(90,155),(91,156),(92,157),(93,158),(94,159),(95,160),(96,153),(97,120),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119)], [(1,11,65,147,155),(2,156,148,66,12),(3,13,67,149,157),(4,158,150,68,14),(5,15,69,151,159),(6,160,152,70,16),(7,9,71,145,153),(8,154,146,72,10),(17,86,94,140,78),(18,79,141,95,87),(19,88,96,142,80),(20,73,143,89,81),(21,82,90,144,74),(22,75,137,91,83),(23,84,92,138,76),(24,77,139,93,85),(25,112,117,50,58),(26,59,51,118,105),(27,106,119,52,60),(28,61,53,120,107),(29,108,113,54,62),(30,63,55,114,109),(31,110,115,56,64),(32,57,49,116,111),(33,100,121,129,45),(34,46,130,122,101),(35,102,123,131,47),(36,48,132,124,103),(37,104,125,133,41),(38,42,134,126,97),(39,98,127,135,43),(40,44,136,128,99)], [(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)]])
C22×C5⋊2C8 is a maximal subgroup of
(C2×C20)⋊8C8 C20.31C42 (C2×C40)⋊15C4 C20.34C42 C10.6M5(2) C5⋊5(C8×D4) C5⋊2C8⋊26D4 C20.35C42 C20.76(C4⋊C4) C42.43D10 C4○D20⋊10C4 C42.47D10 (C2×C10)⋊D8 C5⋊2C8⋊23D4 C5⋊2C8⋊24D4 (C2×C10)⋊Q16 C2×C8×Dic5 C20.51(C4⋊C4) C20.37C42 C4.89(C2×D20) C20.(C2×D4) (D4×C10).24C4 D5×C22×C8
C22×C5⋊2C8 is a maximal quotient of
C42.6Dic5 C40.70C23
64 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 5A | 5B | 8A | ··· | 8P | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D5 | Dic5 | D10 | Dic5 | C5⋊2C8 |
| kernel | C22×C5⋊2C8 | C2×C5⋊2C8 | C22×C20 | C2×C20 | C22×C10 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 6 | 2 | 16 | 2 | 6 | 6 | 2 | 16 |
Matrix representation of C22×C5⋊2C8 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 35 | 40 |
| 0 | 0 | 36 | 40 |
| 27 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 33 | 6 |
| 0 | 0 | 10 | 8 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,35,36,0,0,40,40],[27,0,0,0,0,1,0,0,0,0,33,10,0,0,6,8] >;
C22×C5⋊2C8 in GAP, Magma, Sage, TeX
C_2^2\times C_5\rtimes_2C_8
% in TeX
G:=Group("C2^2xC5:2C8"); // GroupNames label
G:=SmallGroup(160,141);
// by ID
G=gap.SmallGroup(160,141);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations