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, C4, C4, C22, C5, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, Dic10, C2×Dic5, C2×C20, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, C4×Dic10
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, C22×C4, C2×Q8, C4○D4, D10, C4×Q8, Dic10, C4×D5, C22×D5, C2×Dic10, C2×C4×D5, C4○D20, C4×Dic10
►Regular action on 160 points
(1 133 73 52)(2 134 74 53)(3 135 75 54)(4 136 76 55)(5 137 77 56)(6 138 78 57)(7 139 79 58)(8 140 80 59)(9 121 61 60)(10 122 62 41)(11 123 63 42)(12 124 64 43)(13 125 65 44)(14 126 66 45)(15 127 67 46)(16 128 68 47)(17 129 69 48)(18 130 70 49)(19 131 71 50)(20 132 72 51)(21 85 105 159)(22 86 106 160)(23 87 107 141)(24 88 108 142)(25 89 109 143)(26 90 110 144)(27 91 111 145)(28 92 112 146)(29 93 113 147)(30 94 114 148)(31 95 115 149)(32 96 116 150)(33 97 117 151)(34 98 118 152)(35 99 119 153)(36 100 120 154)(37 81 101 155)(38 82 102 156)(39 83 103 157)(40 84 104 158)
(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 35 11 25)(2 34 12 24)(3 33 13 23)(4 32 14 22)(5 31 15 21)(6 30 16 40)(7 29 17 39)(8 28 18 38)(9 27 19 37)(10 26 20 36)(41 144 51 154)(42 143 52 153)(43 142 53 152)(44 141 54 151)(45 160 55 150)(46 159 56 149)(47 158 57 148)(48 157 58 147)(49 156 59 146)(50 155 60 145)(61 111 71 101)(62 110 72 120)(63 109 73 119)(64 108 74 118)(65 107 75 117)(66 106 76 116)(67 105 77 115)(68 104 78 114)(69 103 79 113)(70 102 80 112)(81 121 91 131)(82 140 92 130)(83 139 93 129)(84 138 94 128)(85 137 95 127)(86 136 96 126)(87 135 97 125)(88 134 98 124)(89 133 99 123)(90 132 100 122)
G:=sub<Sym(160)| (1,133,73,52)(2,134,74,53)(3,135,75,54)(4,136,76,55)(5,137,77,56)(6,138,78,57)(7,139,79,58)(8,140,80,59)(9,121,61,60)(10,122,62,41)(11,123,63,42)(12,124,64,43)(13,125,65,44)(14,126,66,45)(15,127,67,46)(16,128,68,47)(17,129,69,48)(18,130,70,49)(19,131,71,50)(20,132,72,51)(21,85,105,159)(22,86,106,160)(23,87,107,141)(24,88,108,142)(25,89,109,143)(26,90,110,144)(27,91,111,145)(28,92,112,146)(29,93,113,147)(30,94,114,148)(31,95,115,149)(32,96,116,150)(33,97,117,151)(34,98,118,152)(35,99,119,153)(36,100,120,154)(37,81,101,155)(38,82,102,156)(39,83,103,157)(40,84,104,158), (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), (1,35,11,25)(2,34,12,24)(3,33,13,23)(4,32,14,22)(5,31,15,21)(6,30,16,40)(7,29,17,39)(8,28,18,38)(9,27,19,37)(10,26,20,36)(41,144,51,154)(42,143,52,153)(43,142,53,152)(44,141,54,151)(45,160,55,150)(46,159,56,149)(47,158,57,148)(48,157,58,147)(49,156,59,146)(50,155,60,145)(61,111,71,101)(62,110,72,120)(63,109,73,119)(64,108,74,118)(65,107,75,117)(66,106,76,116)(67,105,77,115)(68,104,78,114)(69,103,79,113)(70,102,80,112)(81,121,91,131)(82,140,92,130)(83,139,93,129)(84,138,94,128)(85,137,95,127)(86,136,96,126)(87,135,97,125)(88,134,98,124)(89,133,99,123)(90,132,100,122)>;
G:=Group( (1,133,73,52)(2,134,74,53)(3,135,75,54)(4,136,76,55)(5,137,77,56)(6,138,78,57)(7,139,79,58)(8,140,80,59)(9,121,61,60)(10,122,62,41)(11,123,63,42)(12,124,64,43)(13,125,65,44)(14,126,66,45)(15,127,67,46)(16,128,68,47)(17,129,69,48)(18,130,70,49)(19,131,71,50)(20,132,72,51)(21,85,105,159)(22,86,106,160)(23,87,107,141)(24,88,108,142)(25,89,109,143)(26,90,110,144)(27,91,111,145)(28,92,112,146)(29,93,113,147)(30,94,114,148)(31,95,115,149)(32,96,116,150)(33,97,117,151)(34,98,118,152)(35,99,119,153)(36,100,120,154)(37,81,101,155)(38,82,102,156)(39,83,103,157)(40,84,104,158), (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), (1,35,11,25)(2,34,12,24)(3,33,13,23)(4,32,14,22)(5,31,15,21)(6,30,16,40)(7,29,17,39)(8,28,18,38)(9,27,19,37)(10,26,20,36)(41,144,51,154)(42,143,52,153)(43,142,53,152)(44,141,54,151)(45,160,55,150)(46,159,56,149)(47,158,57,148)(48,157,58,147)(49,156,59,146)(50,155,60,145)(61,111,71,101)(62,110,72,120)(63,109,73,119)(64,108,74,118)(65,107,75,117)(66,106,76,116)(67,105,77,115)(68,104,78,114)(69,103,79,113)(70,102,80,112)(81,121,91,131)(82,140,92,130)(83,139,93,129)(84,138,94,128)(85,137,95,127)(86,136,96,126)(87,135,97,125)(88,134,98,124)(89,133,99,123)(90,132,100,122) );
G=PermutationGroup([[(1,133,73,52),(2,134,74,53),(3,135,75,54),(4,136,76,55),(5,137,77,56),(6,138,78,57),(7,139,79,58),(8,140,80,59),(9,121,61,60),(10,122,62,41),(11,123,63,42),(12,124,64,43),(13,125,65,44),(14,126,66,45),(15,127,67,46),(16,128,68,47),(17,129,69,48),(18,130,70,49),(19,131,71,50),(20,132,72,51),(21,85,105,159),(22,86,106,160),(23,87,107,141),(24,88,108,142),(25,89,109,143),(26,90,110,144),(27,91,111,145),(28,92,112,146),(29,93,113,147),(30,94,114,148),(31,95,115,149),(32,96,116,150),(33,97,117,151),(34,98,118,152),(35,99,119,153),(36,100,120,154),(37,81,101,155),(38,82,102,156),(39,83,103,157),(40,84,104,158)], [(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)], [(1,35,11,25),(2,34,12,24),(3,33,13,23),(4,32,14,22),(5,31,15,21),(6,30,16,40),(7,29,17,39),(8,28,18,38),(9,27,19,37),(10,26,20,36),(41,144,51,154),(42,143,52,153),(43,142,53,152),(44,141,54,151),(45,160,55,150),(46,159,56,149),(47,158,57,148),(48,157,58,147),(49,156,59,146),(50,155,60,145),(61,111,71,101),(62,110,72,120),(63,109,73,119),(64,108,74,118),(65,107,75,117),(66,106,76,116),(67,105,77,115),(68,104,78,114),(69,103,79,113),(70,102,80,112),(81,121,91,131),(82,140,92,130),(83,139,93,129),(84,138,94,128),(85,137,95,127),(86,136,96,126),(87,135,97,125),(88,134,98,124),(89,133,99,123),(90,132,100,122)]])
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