direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Dic10, C20⋊3Q8, C42.3D5, C5⋊2(C4×Q8), C4.9(C4×D5), (C4×C20).5C2, C10.1(C2×Q8), C20.40(C2×C4), (C2×C4).72D10, C10.1(C4○D4), C2.1(C4○D20), C4⋊Dic5.13C2, (C2×C10).9C23, Dic5.3(C2×C4), (C4×Dic5).9C2, C2.1(C2×Dic10), C10.14(C22×C4), (C2×C20).84C22, C10.D4.7C2, C22.8(C22×D5), (C2×Dic10).11C2, (C2×Dic5).26C22, C2.4(C2×C4×D5), SmallGroup(160,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Dic10
G = < a,b,c | a4=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 168 in 70 conjugacy classes, 45 normal (21 characteristic)
C1, C2 [×3], C4 [×4], C4 [×7], C22, C5, C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], C42, C42 [×2], C4⋊C4 [×3], C2×Q8, Dic5 [×4], Dic5 [×2], C20 [×4], C20, C2×C10, C4×Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20 [×3], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4×C20, C2×Dic10, C4×Dic10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, D5, C22×C4, C2×Q8, C4○D4, D10 [×3], C4×Q8, Dic10 [×2], C4×D5 [×2], C22×D5, C2×Dic10, C2×C4×D5, C4○D20, C4×Dic10
►Regular action on 160 points
(1 59 79 120)(2 60 80 101)(3 41 61 102)(4 42 62 103)(5 43 63 104)(6 44 64 105)(7 45 65 106)(8 46 66 107)(9 47 67 108)(10 48 68 109)(11 49 69 110)(12 50 70 111)(13 51 71 112)(14 52 72 113)(15 53 73 114)(16 54 74 115)(17 55 75 116)(18 56 76 117)(19 57 77 118)(20 58 78 119)(21 127 89 151)(22 128 90 152)(23 129 91 153)(24 130 92 154)(25 131 93 155)(26 132 94 156)(27 133 95 157)(28 134 96 158)(29 135 97 159)(30 136 98 160)(31 137 99 141)(32 138 100 142)(33 139 81 143)(34 140 82 144)(35 121 83 145)(36 122 84 146)(37 123 85 147)(38 124 86 148)(39 125 87 149)(40 126 88 150)
(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)
(1 141 11 151)(2 160 12 150)(3 159 13 149)(4 158 14 148)(5 157 15 147)(6 156 16 146)(7 155 17 145)(8 154 18 144)(9 153 19 143)(10 152 20 142)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)(61 135 71 125)(62 134 72 124)(63 133 73 123)(64 132 74 122)(65 131 75 121)(66 130 76 140)(67 129 77 139)(68 128 78 138)(69 127 79 137)(70 126 80 136)(81 108 91 118)(82 107 92 117)(83 106 93 116)(84 105 94 115)(85 104 95 114)(86 103 96 113)(87 102 97 112)(88 101 98 111)(89 120 99 110)(90 119 100 109)
G:=sub<Sym(160)| (1,59,79,120)(2,60,80,101)(3,41,61,102)(4,42,62,103)(5,43,63,104)(6,44,64,105)(7,45,65,106)(8,46,66,107)(9,47,67,108)(10,48,68,109)(11,49,69,110)(12,50,70,111)(13,51,71,112)(14,52,72,113)(15,53,73,114)(16,54,74,115)(17,55,75,116)(18,56,76,117)(19,57,77,118)(20,58,78,119)(21,127,89,151)(22,128,90,152)(23,129,91,153)(24,130,92,154)(25,131,93,155)(26,132,94,156)(27,133,95,157)(28,134,96,158)(29,135,97,159)(30,136,98,160)(31,137,99,141)(32,138,100,142)(33,139,81,143)(34,140,82,144)(35,121,83,145)(36,122,84,146)(37,123,85,147)(38,124,86,148)(39,125,87,149)(40,126,88,150), (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), (1,141,11,151)(2,160,12,150)(3,159,13,149)(4,158,14,148)(5,157,15,147)(6,156,16,146)(7,155,17,145)(8,154,18,144)(9,153,19,143)(10,152,20,142)(21,59,31,49)(22,58,32,48)(23,57,33,47)(24,56,34,46)(25,55,35,45)(26,54,36,44)(27,53,37,43)(28,52,38,42)(29,51,39,41)(30,50,40,60)(61,135,71,125)(62,134,72,124)(63,133,73,123)(64,132,74,122)(65,131,75,121)(66,130,76,140)(67,129,77,139)(68,128,78,138)(69,127,79,137)(70,126,80,136)(81,108,91,118)(82,107,92,117)(83,106,93,116)(84,105,94,115)(85,104,95,114)(86,103,96,113)(87,102,97,112)(88,101,98,111)(89,120,99,110)(90,119,100,109)>;
G:=Group( (1,59,79,120)(2,60,80,101)(3,41,61,102)(4,42,62,103)(5,43,63,104)(6,44,64,105)(7,45,65,106)(8,46,66,107)(9,47,67,108)(10,48,68,109)(11,49,69,110)(12,50,70,111)(13,51,71,112)(14,52,72,113)(15,53,73,114)(16,54,74,115)(17,55,75,116)(18,56,76,117)(19,57,77,118)(20,58,78,119)(21,127,89,151)(22,128,90,152)(23,129,91,153)(24,130,92,154)(25,131,93,155)(26,132,94,156)(27,133,95,157)(28,134,96,158)(29,135,97,159)(30,136,98,160)(31,137,99,141)(32,138,100,142)(33,139,81,143)(34,140,82,144)(35,121,83,145)(36,122,84,146)(37,123,85,147)(38,124,86,148)(39,125,87,149)(40,126,88,150), (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), (1,141,11,151)(2,160,12,150)(3,159,13,149)(4,158,14,148)(5,157,15,147)(6,156,16,146)(7,155,17,145)(8,154,18,144)(9,153,19,143)(10,152,20,142)(21,59,31,49)(22,58,32,48)(23,57,33,47)(24,56,34,46)(25,55,35,45)(26,54,36,44)(27,53,37,43)(28,52,38,42)(29,51,39,41)(30,50,40,60)(61,135,71,125)(62,134,72,124)(63,133,73,123)(64,132,74,122)(65,131,75,121)(66,130,76,140)(67,129,77,139)(68,128,78,138)(69,127,79,137)(70,126,80,136)(81,108,91,118)(82,107,92,117)(83,106,93,116)(84,105,94,115)(85,104,95,114)(86,103,96,113)(87,102,97,112)(88,101,98,111)(89,120,99,110)(90,119,100,109) );
G=PermutationGroup([(1,59,79,120),(2,60,80,101),(3,41,61,102),(4,42,62,103),(5,43,63,104),(6,44,64,105),(7,45,65,106),(8,46,66,107),(9,47,67,108),(10,48,68,109),(11,49,69,110),(12,50,70,111),(13,51,71,112),(14,52,72,113),(15,53,73,114),(16,54,74,115),(17,55,75,116),(18,56,76,117),(19,57,77,118),(20,58,78,119),(21,127,89,151),(22,128,90,152),(23,129,91,153),(24,130,92,154),(25,131,93,155),(26,132,94,156),(27,133,95,157),(28,134,96,158),(29,135,97,159),(30,136,98,160),(31,137,99,141),(32,138,100,142),(33,139,81,143),(34,140,82,144),(35,121,83,145),(36,122,84,146),(37,123,85,147),(38,124,86,148),(39,125,87,149),(40,126,88,150)], [(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)], [(1,141,11,151),(2,160,12,150),(3,159,13,149),(4,158,14,148),(5,157,15,147),(6,156,16,146),(7,155,17,145),(8,154,18,144),(9,153,19,143),(10,152,20,142),(21,59,31,49),(22,58,32,48),(23,57,33,47),(24,56,34,46),(25,55,35,45),(26,54,36,44),(27,53,37,43),(28,52,38,42),(29,51,39,41),(30,50,40,60),(61,135,71,125),(62,134,72,124),(63,133,73,123),(64,132,74,122),(65,131,75,121),(66,130,76,140),(67,129,77,139),(68,128,78,138),(69,127,79,137),(70,126,80,136),(81,108,91,118),(82,107,92,117),(83,106,93,116),(84,105,94,115),(85,104,95,114),(86,103,96,113),(87,102,97,112),(88,101,98,111),(89,120,99,110),(90,119,100,109)])
C4×Dic10 is a maximal subgroup of
Dic10⋊3C8 Dic10⋊4C8 C40⋊11Q8 C40⋊Q8 C42.16D10 Dic20⋊9C4 Dic5.5M4(2) Dic10.3Q8 Dic10⋊5C8 C42.198D10 C42.36D10 Dic10⋊8D4 C4⋊Dic20 C20.7Q16 Dic10⋊4Q8 C42.51D10 C42.59D10 C42.61D10 Dic10.4Q8 Dic10⋊9D4 C20⋊Q16 Dic10⋊5Q8 Dic10⋊6Q8 C42.274D10 C42.277D10 C42.87D10 C42.88D10 C42.89D10 C42.91D10 C42.93D10 C42.96D10 C42.98D10 C42.99D10 C42.102D10 D4⋊5Dic10 C42.105D10 C42.106D10 D4⋊6Dic10 C42.108D10 Dic10⋊23D4 Dic10⋊24D4 C42.229D10 C42.114D10 C42.115D10 Dic10⋊10Q8 C42.122D10 Q8⋊5Dic10 Q8⋊6Dic10 C4×Q8×D5 C42.125D10 C42.232D10 C42.134D10 C42.135D10 C42.136D10 C42.137D10 C42.139D10 Dic10⋊10D4 C42.143D10 Dic10⋊7Q8 D20⋊7Q8 C42.152D10 C42.154D10 C42.159D10 C42.160D10 C42.162D10 C42.164D10 C42.166D10 Dic10⋊11D4 Dic10⋊8Q8 Dic10⋊9Q8 D20⋊8Q8 D20⋊9Q8 C42.177D10 Dic3⋊5Dic10 Dic30⋊14C4
C4×Dic10 is a maximal quotient of
(C2×C20)⋊Q8 C10.49(C4×D4) C4⋊Dic5⋊15C4 C10.52(C4×D4) C40⋊11Q8 C40⋊Q8 C20⋊7(C4⋊C4) (C2×C20)⋊10Q8 C10.92(C4×D4) Dic3⋊5Dic10 Dic30⋊14C4
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D5 | C4○D4 | D10 | Dic10 | C4×D5 | C4○D20 |
| kernel | C4×Dic10 | C4×Dic5 | C10.D4 | C4⋊Dic5 | C4×C20 | C2×Dic10 | Dic10 | C20 | C42 | C10 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 6 | 8 | 8 | 8 |
Matrix representation of C4×Dic10 ►in GL3(𝔽41) generated by
| 32 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 40 | 0 | 0 |
| 0 | 9 | 11 |
| 0 | 30 | 14 |
| 40 | 0 | 0 |
| 0 | 27 | 12 |
| 0 | 28 | 14 |
G:=sub<GL(3,GF(41))| [32,0,0,0,9,0,0,0,9],[40,0,0,0,9,30,0,11,14],[40,0,0,0,27,28,0,12,14] >;
C4×Dic10 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{10} % in TeX
G:=Group("C4xDic10"); // GroupNames label
G:=SmallGroup(160,89);
// by ID
G=gap.SmallGroup(160,89);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,103,50,4613]);
// Polycyclic
G:=Group<a,b,c|a^4=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations