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, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C10.D4, C22×Dic5, C22×C20, C2×C10.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, Dic10, C4×D5, C5⋊D4, C22×D5, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4
►Regular action on 160 points
Generators in S160
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(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)(91 150)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 131)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(129 160)(130 151)
(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 96)(2 109 26 95)(3 108 27 94)(4 107 28 93)(5 106 29 92)(6 105 30 91)(7 104 21 100)(8 103 22 99)(9 102 23 98)(10 101 24 97)(11 90 160 76)(12 89 151 75)(13 88 152 74)(14 87 153 73)(15 86 154 72)(16 85 155 71)(17 84 156 80)(18 83 157 79)(19 82 158 78)(20 81 159 77)(31 129 46 115)(32 128 47 114)(33 127 48 113)(34 126 49 112)(35 125 50 111)(36 124 41 120)(37 123 42 119)(38 122 43 118)(39 121 44 117)(40 130 45 116)(51 149 66 135)(52 148 67 134)(53 147 68 133)(54 146 69 132)(55 145 70 131)(56 144 61 140)(57 143 62 139)(58 142 63 138)(59 141 64 137)(60 150 65 136)
(1 111 6 116)(2 120 7 115)(3 119 8 114)(4 118 9 113)(5 117 10 112)(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 95 36 100)(32 94 37 99)(33 93 38 98)(34 92 39 97)(35 91 40 96)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 160 56 155)(52 159 57 154)(53 158 58 153)(54 157 59 152)(55 156 60 151)(71 135 76 140)(72 134 77 139)(73 133 78 138)(74 132 79 137)(75 131 80 136)(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)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,111)(18,112)(19,113)(20,114)(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)(91,150)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,131)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,151), (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), (1,110,25,96)(2,109,26,95)(3,108,27,94)(4,107,28,93)(5,106,29,92)(6,105,30,91)(7,104,21,100)(8,103,22,99)(9,102,23,98)(10,101,24,97)(11,90,160,76)(12,89,151,75)(13,88,152,74)(14,87,153,73)(15,86,154,72)(16,85,155,71)(17,84,156,80)(18,83,157,79)(19,82,158,78)(20,81,159,77)(31,129,46,115)(32,128,47,114)(33,127,48,113)(34,126,49,112)(35,125,50,111)(36,124,41,120)(37,123,42,119)(38,122,43,118)(39,121,44,117)(40,130,45,116)(51,149,66,135)(52,148,67,134)(53,147,68,133)(54,146,69,132)(55,145,70,131)(56,144,61,140)(57,143,62,139)(58,142,63,138)(59,141,64,137)(60,150,65,136), (1,111,6,116)(2,120,7,115)(3,119,8,114)(4,118,9,113)(5,117,10,112)(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)(31,95,36,100)(32,94,37,99)(33,93,38,98)(34,92,39,97)(35,91,40,96)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,160,56,155)(52,159,57,154)(53,158,58,153)(54,157,59,152)(55,156,60,151)(71,135,76,140)(72,134,77,139)(73,133,78,138)(74,132,79,137)(75,131,80,136)(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)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,111)(18,112)(19,113)(20,114)(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)(91,150)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,131)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,151), (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), (1,110,25,96)(2,109,26,95)(3,108,27,94)(4,107,28,93)(5,106,29,92)(6,105,30,91)(7,104,21,100)(8,103,22,99)(9,102,23,98)(10,101,24,97)(11,90,160,76)(12,89,151,75)(13,88,152,74)(14,87,153,73)(15,86,154,72)(16,85,155,71)(17,84,156,80)(18,83,157,79)(19,82,158,78)(20,81,159,77)(31,129,46,115)(32,128,47,114)(33,127,48,113)(34,126,49,112)(35,125,50,111)(36,124,41,120)(37,123,42,119)(38,122,43,118)(39,121,44,117)(40,130,45,116)(51,149,66,135)(52,148,67,134)(53,147,68,133)(54,146,69,132)(55,145,70,131)(56,144,61,140)(57,143,62,139)(58,142,63,138)(59,141,64,137)(60,150,65,136), (1,111,6,116)(2,120,7,115)(3,119,8,114)(4,118,9,113)(5,117,10,112)(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)(31,95,36,100)(32,94,37,99)(33,93,38,98)(34,92,39,97)(35,91,40,96)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(51,160,56,155)(52,159,57,154)(53,158,58,153)(54,157,59,152)(55,156,60,151)(71,135,76,140)(72,134,77,139)(73,133,78,138)(74,132,79,137)(75,131,80,136)(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),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,111),(18,112),(19,113),(20,114),(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),(91,150),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149),(101,132),(102,133),(103,134),(104,135),(105,136),(106,137),(107,138),(108,139),(109,140),(110,131),(121,152),(122,153),(123,154),(124,155),(125,156),(126,157),(127,158),(128,159),(129,160),(130,151)], [(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)], [(1,110,25,96),(2,109,26,95),(3,108,27,94),(4,107,28,93),(5,106,29,92),(6,105,30,91),(7,104,21,100),(8,103,22,99),(9,102,23,98),(10,101,24,97),(11,90,160,76),(12,89,151,75),(13,88,152,74),(14,87,153,73),(15,86,154,72),(16,85,155,71),(17,84,156,80),(18,83,157,79),(19,82,158,78),(20,81,159,77),(31,129,46,115),(32,128,47,114),(33,127,48,113),(34,126,49,112),(35,125,50,111),(36,124,41,120),(37,123,42,119),(38,122,43,118),(39,121,44,117),(40,130,45,116),(51,149,66,135),(52,148,67,134),(53,147,68,133),(54,146,69,132),(55,145,70,131),(56,144,61,140),(57,143,62,139),(58,142,63,138),(59,141,64,137),(60,150,65,136)], [(1,111,6,116),(2,120,7,115),(3,119,8,114),(4,118,9,113),(5,117,10,112),(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),(31,95,36,100),(32,94,37,99),(33,93,38,98),(34,92,39,97),(35,91,40,96),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(51,160,56,155),(52,159,57,154),(53,158,58,153),(54,157,59,152),(55,156,60,151),(71,135,76,140),(72,134,77,139),(73,133,78,138),(74,132,79,137),(75,131,80,136),(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