direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C10.D4, C23.29D10, C22.4Dic10, C10⋊2(C4⋊C4), C10.7(C2×Q8), (C2×C10).5Q8, (C2×Dic5)⋊5C4, Dic5⋊5(C2×C4), (C2×C4).64D10, (C2×C10).35D4, C10.38(C2×D4), (C22×C4).3D5, (C22×C20).4C2, C2.2(C2×Dic10), C22.16(C4×D5), C10.30(C22×C4), (C2×C10).41C23, (C2×C20).76C22, C22.19(C5⋊D4), (C22×Dic5).4C2, C22.20(C22×D5), (C22×C10).33C22, (C2×Dic5).36C22, C5⋊3(C2×C4⋊C4), C2.18(C2×C4×D5), C2.1(C2×C5⋊D4), (C2×C10).37(C2×C4), SmallGroup(160,144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C10.D4
G = < a,b,c,d | a2=b10=c4=1, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >
Subgroups: 216 in 92 conjugacy classes, 57 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C22, C22 [×6], C5, C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×6], C2×C4⋊C4, C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C10.D4 [×4], C22×Dic5 [×2], C22×C20, C2×C10.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4
►Regular action on 160 points
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 119)(12 120)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 90)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(91 146)(92 147)(93 148)(94 149)(95 150)(96 141)(97 142)(98 143)(99 144)(100 145)(101 136)(102 137)(103 138)(104 139)(105 140)(106 131)(107 132)(108 133)(109 134)(110 135)(121 156)(122 157)(123 158)(124 159)(125 160)(126 151)(127 152)(128 153)(129 154)(130 155)
(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)
(1 110 25 100)(2 109 26 99)(3 108 27 98)(4 107 28 97)(5 106 29 96)(6 105 30 95)(7 104 21 94)(8 103 22 93)(9 102 23 92)(10 101 24 91)(11 90 154 76)(12 89 155 75)(13 88 156 74)(14 87 157 73)(15 86 158 72)(16 85 159 71)(17 84 160 80)(18 83 151 79)(19 82 152 78)(20 81 153 77)(31 129 46 119)(32 128 47 118)(33 127 48 117)(34 126 49 116)(35 125 50 115)(36 124 41 114)(37 123 42 113)(38 122 43 112)(39 121 44 111)(40 130 45 120)(51 149 66 139)(52 148 67 138)(53 147 68 137)(54 146 69 136)(55 145 70 135)(56 144 61 134)(57 143 62 133)(58 142 63 132)(59 141 64 131)(60 150 65 140)
(1 115 6 120)(2 114 7 119)(3 113 8 118)(4 112 9 117)(5 111 10 116)(11 61 16 66)(12 70 17 65)(13 69 18 64)(14 68 19 63)(15 67 20 62)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 154 56 159)(52 153 57 158)(53 152 58 157)(54 151 59 156)(55 160 60 155)(71 139 76 134)(72 138 77 133)(73 137 78 132)(74 136 79 131)(75 135 80 140)(81 143 86 148)(82 142 87 147)(83 141 88 146)(84 150 89 145)(85 149 90 144)
G:=sub<Sym(160)| (1,70)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,119)(12,120)(13,111)(14,112)(15,113)(16,114)(17,115)(18,116)(19,117)(20,118)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,90)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(91,146)(92,147)(93,148)(94,149)(95,150)(96,141)(97,142)(98,143)(99,144)(100,145)(101,136)(102,137)(103,138)(104,139)(105,140)(106,131)(107,132)(108,133)(109,134)(110,135)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155), (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), (1,110,25,100)(2,109,26,99)(3,108,27,98)(4,107,28,97)(5,106,29,96)(6,105,30,95)(7,104,21,94)(8,103,22,93)(9,102,23,92)(10,101,24,91)(11,90,154,76)(12,89,155,75)(13,88,156,74)(14,87,157,73)(15,86,158,72)(16,85,159,71)(17,84,160,80)(18,83,151,79)(19,82,152,78)(20,81,153,77)(31,129,46,119)(32,128,47,118)(33,127,48,117)(34,126,49,116)(35,125,50,115)(36,124,41,114)(37,123,42,113)(38,122,43,112)(39,121,44,111)(40,130,45,120)(51,149,66,139)(52,148,67,138)(53,147,68,137)(54,146,69,136)(55,145,70,135)(56,144,61,134)(57,143,62,133)(58,142,63,132)(59,141,64,131)(60,150,65,140), (1,115,6,120)(2,114,7,119)(3,113,8,118)(4,112,9,117)(5,111,10,116)(11,61,16,66)(12,70,17,65)(13,69,18,64)(14,68,19,63)(15,67,20,62)(21,129,26,124)(22,128,27,123)(23,127,28,122)(24,126,29,121)(25,125,30,130)(31,99,36,94)(32,98,37,93)(33,97,38,92)(34,96,39,91)(35,95,40,100)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,154,56,159)(52,153,57,158)(53,152,58,157)(54,151,59,156)(55,160,60,155)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,143,86,148)(82,142,87,147)(83,141,88,146)(84,150,89,145)(85,149,90,144)>;
G:=Group( (1,70)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,119)(12,120)(13,111)(14,112)(15,113)(16,114)(17,115)(18,116)(19,117)(20,118)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,90)(32,81)(33,82)(34,83)(35,84)(36,85)(37,86)(38,87)(39,88)(40,89)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(91,146)(92,147)(93,148)(94,149)(95,150)(96,141)(97,142)(98,143)(99,144)(100,145)(101,136)(102,137)(103,138)(104,139)(105,140)(106,131)(107,132)(108,133)(109,134)(110,135)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155), (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), (1,110,25,100)(2,109,26,99)(3,108,27,98)(4,107,28,97)(5,106,29,96)(6,105,30,95)(7,104,21,94)(8,103,22,93)(9,102,23,92)(10,101,24,91)(11,90,154,76)(12,89,155,75)(13,88,156,74)(14,87,157,73)(15,86,158,72)(16,85,159,71)(17,84,160,80)(18,83,151,79)(19,82,152,78)(20,81,153,77)(31,129,46,119)(32,128,47,118)(33,127,48,117)(34,126,49,116)(35,125,50,115)(36,124,41,114)(37,123,42,113)(38,122,43,112)(39,121,44,111)(40,130,45,120)(51,149,66,139)(52,148,67,138)(53,147,68,137)(54,146,69,136)(55,145,70,135)(56,144,61,134)(57,143,62,133)(58,142,63,132)(59,141,64,131)(60,150,65,140), (1,115,6,120)(2,114,7,119)(3,113,8,118)(4,112,9,117)(5,111,10,116)(11,61,16,66)(12,70,17,65)(13,69,18,64)(14,68,19,63)(15,67,20,62)(21,129,26,124)(22,128,27,123)(23,127,28,122)(24,126,29,121)(25,125,30,130)(31,99,36,94)(32,98,37,93)(33,97,38,92)(34,96,39,91)(35,95,40,100)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,154,56,159)(52,153,57,158)(53,152,58,157)(54,151,59,156)(55,160,60,155)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,143,86,148)(82,142,87,147)(83,141,88,146)(84,150,89,145)(85,149,90,144) );
G=PermutationGroup([(1,70),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,119),(12,120),(13,111),(14,112),(15,113),(16,114),(17,115),(18,116),(19,117),(20,118),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,90),(32,81),(33,82),(34,83),(35,84),(36,85),(37,86),(38,87),(39,88),(40,89),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(91,146),(92,147),(93,148),(94,149),(95,150),(96,141),(97,142),(98,143),(99,144),(100,145),(101,136),(102,137),(103,138),(104,139),(105,140),(106,131),(107,132),(108,133),(109,134),(110,135),(121,156),(122,157),(123,158),(124,159),(125,160),(126,151),(127,152),(128,153),(129,154),(130,155)], [(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)], [(1,110,25,100),(2,109,26,99),(3,108,27,98),(4,107,28,97),(5,106,29,96),(6,105,30,95),(7,104,21,94),(8,103,22,93),(9,102,23,92),(10,101,24,91),(11,90,154,76),(12,89,155,75),(13,88,156,74),(14,87,157,73),(15,86,158,72),(16,85,159,71),(17,84,160,80),(18,83,151,79),(19,82,152,78),(20,81,153,77),(31,129,46,119),(32,128,47,118),(33,127,48,117),(34,126,49,116),(35,125,50,115),(36,124,41,114),(37,123,42,113),(38,122,43,112),(39,121,44,111),(40,130,45,120),(51,149,66,139),(52,148,67,138),(53,147,68,137),(54,146,69,136),(55,145,70,135),(56,144,61,134),(57,143,62,133),(58,142,63,132),(59,141,64,131),(60,150,65,140)], [(1,115,6,120),(2,114,7,119),(3,113,8,118),(4,112,9,117),(5,111,10,116),(11,61,16,66),(12,70,17,65),(13,69,18,64),(14,68,19,63),(15,67,20,62),(21,129,26,124),(22,128,27,123),(23,127,28,122),(24,126,29,121),(25,125,30,130),(31,99,36,94),(32,98,37,93),(33,97,38,92),(34,96,39,91),(35,95,40,100),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(51,154,56,159),(52,153,57,158),(53,152,58,157),(54,151,59,156),(55,160,60,155),(71,139,76,134),(72,138,77,133),(73,137,78,132),(74,136,79,131),(75,135,80,140),(81,143,86,148),(82,142,87,147),(83,141,88,146),(84,150,89,145),(85,149,90,144)])
C2×C10.D4 is a maximal subgroup of
(C2×Dic5)⋊C8 (C22×C4).F5 C22.F5⋊C4 (C2×C20)⋊Q8 C10.49(C4×D4) Dic5⋊2C42 C5⋊2(C42⋊8C4) C10.51(C4×D4) C2.(C4×D20) C4⋊Dic5⋊15C4 C10.52(C4×D4) (C2×Dic5)⋊Q8 C2.(C20⋊Q8) (C2×Dic5).Q8 (C2×C20).28D4 (C2×C4).Dic10 C10.(C4⋊Q8) D10⋊2(C4⋊C4) D10⋊3(C4⋊C4) C10.54(C4×D4) (C2×Dic5)⋊3D4 C10.(C4⋊D4) (C22×D5).Q8 C20⋊7(C4⋊C4) C10.92(C4×D4) (C2×C42)⋊D5 C24.44D10 C24.3D10 C24.4D10 C24.46D10 C24.6D10 C24.7D10 C24.9D10 C24.13D10 C24.14D10 C10.96(C4×D4) C20⋊4(C4⋊C4) C20⋊5(C4⋊C4) C10.97(C4×D4) (C2×C4)⋊Dic10 (C2×C20).287D4 (C2×C20).288D4 (C2×C20).53D4 (C2×C20).54D4 D10⋊5(C4⋊C4) (C2×C20).289D4 (C2×C20).56D4 C24.62D10 C24.20D10 C10.C22≀C2 C2×C4×Dic10 C2×D5×C4⋊C4 C42.96D10 D4⋊5Dic10 C42.104D10 C42.108D10 C42.118D10 C10.682- 1+4 C10.342+ 1+4 C10.352+ 1+4 C10.502+ 1+4 C10.522+ 1+4 C10.572+ 1+4 C10.802- 1+4 C10.822- 1+4 C2×C4×C5⋊D4 C10.1042- 1+4
C2×C10.D4 is a maximal quotient of
C20⋊7(C4⋊C4) C24.44D10 C24.46D10 C20.47(C4⋊C4) C20⋊4(C4⋊C4) C20⋊5(C4⋊C4) C20.48(C4⋊C4) C20.64(C4⋊C4) C20.76(C4⋊C4) C20.65(C4⋊C4) Dic5⋊5M4(2) C20.51(C4⋊C4) C23.Dic10 C24.62D10
52 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D5 | D10 | D10 | Dic10 | C4×D5 | C5⋊D4 |
| kernel | C2×C10.D4 | C10.D4 | C22×Dic5 | C22×C20 | C2×Dic5 | C2×C10 | C2×C10 | C22×C4 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 8 |
Matrix representation of C2×C10.D4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 6 |
| 0 | 0 | 34 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 24 | 23 |
| 0 | 0 | 7 | 17 |
| 9 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,34,34,0,0,6,0],[1,0,0,0,0,1,0,0,0,0,24,7,0,0,23,17],[9,0,0,0,0,1,0,0,0,0,1,0,0,0,40,40] >;
C2×C10.D4 in GAP, Magma, Sage, TeX
C_2\times C_{10}.D_4 % in TeX
G:=Group("C2xC10.D4"); // GroupNames label
G:=SmallGroup(160,144);
// by ID
G=gap.SmallGroup(160,144);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,362,50,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^4=1,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations