metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊3Q8, Dic10⋊8C4, C5⋊3(C4×Q8), C4⋊C4.7D5, C4.4(C4×D5), C2.1(Q8×D5), C20.31(C2×C4), (C2×C4).29D10, C10.10(C2×Q8), Dic5.4(C2×C4), C10.24(C4○D4), C2.3(D4⋊2D5), C10.21(C22×C4), (C2×C20).21C22, (C2×C10).28C23, (C4×Dic5).10C2, (C2×Dic10).8C2, C10.D4.4C2, C22.15(C22×D5), (C2×Dic5).62C22, C2.10(C2×C4×D5), (C5×C4⋊C4).4C2, SmallGroup(160,108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊3Q8
G = < a,b,c,d | a10=c4=1, b2=a5, d2=c2, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 168 in 70 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic5, C4×Dic5, C10.D4, C5×C4⋊C4, C2×Dic10, Dic5⋊3Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, C22×C4, C2×Q8, C4○D4, D10, C4×Q8, C4×D5, C22×D5, C2×C4×D5, D4⋊2D5, Q8×D5, Dic5⋊3Q8
►Regular action on 160 points
Generators in S160
(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 96 6 91)(2 95 7 100)(3 94 8 99)(4 93 9 98)(5 92 10 97)(11 84 16 89)(12 83 17 88)(13 82 18 87)(14 81 19 86)(15 90 20 85)(21 114 26 119)(22 113 27 118)(23 112 28 117)(24 111 29 116)(25 120 30 115)(31 104 36 109)(32 103 37 108)(33 102 38 107)(34 101 39 106)(35 110 40 105)(41 134 46 139)(42 133 47 138)(43 132 48 137)(44 131 49 136)(45 140 50 135)(51 124 56 129)(52 123 57 128)(53 122 58 127)(54 121 59 126)(55 130 60 125)(61 154 66 159)(62 153 67 158)(63 152 68 157)(64 151 69 156)(65 160 70 155)(71 144 76 149)(72 143 77 148)(73 142 78 147)(74 141 79 146)(75 150 80 145)
(1 101 19 111)(2 110 20 120)(3 109 11 119)(4 108 12 118)(5 107 13 117)(6 106 14 116)(7 105 15 115)(8 104 16 114)(9 103 17 113)(10 102 18 112)(21 94 31 84)(22 93 32 83)(23 92 33 82)(24 91 34 81)(25 100 35 90)(26 99 36 89)(27 98 37 88)(28 97 38 87)(29 96 39 86)(30 95 40 85)(41 159 51 149)(42 158 52 148)(43 157 53 147)(44 156 54 146)(45 155 55 145)(46 154 56 144)(47 153 57 143)(48 152 58 142)(49 151 59 141)(50 160 60 150)(61 124 71 134)(62 123 72 133)(63 122 73 132)(64 121 74 131)(65 130 75 140)(66 129 76 139)(67 128 77 138)(68 127 78 137)(69 126 79 136)(70 125 80 135)
(1 59 19 49)(2 60 20 50)(3 51 11 41)(4 52 12 42)(5 53 13 43)(6 54 14 44)(7 55 15 45)(8 56 16 46)(9 57 17 47)(10 58 18 48)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)
G:=sub<Sym(160)| (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), (1,96,6,91)(2,95,7,100)(3,94,8,99)(4,93,9,98)(5,92,10,97)(11,84,16,89)(12,83,17,88)(13,82,18,87)(14,81,19,86)(15,90,20,85)(21,114,26,119)(22,113,27,118)(23,112,28,117)(24,111,29,116)(25,120,30,115)(31,104,36,109)(32,103,37,108)(33,102,38,107)(34,101,39,106)(35,110,40,105)(41,134,46,139)(42,133,47,138)(43,132,48,137)(44,131,49,136)(45,140,50,135)(51,124,56,129)(52,123,57,128)(53,122,58,127)(54,121,59,126)(55,130,60,125)(61,154,66,159)(62,153,67,158)(63,152,68,157)(64,151,69,156)(65,160,70,155)(71,144,76,149)(72,143,77,148)(73,142,78,147)(74,141,79,146)(75,150,80,145), (1,101,19,111)(2,110,20,120)(3,109,11,119)(4,108,12,118)(5,107,13,117)(6,106,14,116)(7,105,15,115)(8,104,16,114)(9,103,17,113)(10,102,18,112)(21,94,31,84)(22,93,32,83)(23,92,33,82)(24,91,34,81)(25,100,35,90)(26,99,36,89)(27,98,37,88)(28,97,38,87)(29,96,39,86)(30,95,40,85)(41,159,51,149)(42,158,52,148)(43,157,53,147)(44,156,54,146)(45,155,55,145)(46,154,56,144)(47,153,57,143)(48,152,58,142)(49,151,59,141)(50,160,60,150)(61,124,71,134)(62,123,72,133)(63,122,73,132)(64,121,74,131)(65,130,75,140)(66,129,76,139)(67,128,77,138)(68,127,78,137)(69,126,79,136)(70,125,80,135), (1,59,19,49)(2,60,20,50)(3,51,11,41)(4,52,12,42)(5,53,13,43)(6,54,14,44)(7,55,15,45)(8,56,16,46)(9,57,17,47)(10,58,18,48)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,131,91,121)(82,132,92,122)(83,133,93,123)(84,134,94,124)(85,135,95,125)(86,136,96,126)(87,137,97,127)(88,138,98,128)(89,139,99,129)(90,140,100,130)(101,151,111,141)(102,152,112,142)(103,153,113,143)(104,154,114,144)(105,155,115,145)(106,156,116,146)(107,157,117,147)(108,158,118,148)(109,159,119,149)(110,160,120,150)>;
G:=Group( (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), (1,96,6,91)(2,95,7,100)(3,94,8,99)(4,93,9,98)(5,92,10,97)(11,84,16,89)(12,83,17,88)(13,82,18,87)(14,81,19,86)(15,90,20,85)(21,114,26,119)(22,113,27,118)(23,112,28,117)(24,111,29,116)(25,120,30,115)(31,104,36,109)(32,103,37,108)(33,102,38,107)(34,101,39,106)(35,110,40,105)(41,134,46,139)(42,133,47,138)(43,132,48,137)(44,131,49,136)(45,140,50,135)(51,124,56,129)(52,123,57,128)(53,122,58,127)(54,121,59,126)(55,130,60,125)(61,154,66,159)(62,153,67,158)(63,152,68,157)(64,151,69,156)(65,160,70,155)(71,144,76,149)(72,143,77,148)(73,142,78,147)(74,141,79,146)(75,150,80,145), (1,101,19,111)(2,110,20,120)(3,109,11,119)(4,108,12,118)(5,107,13,117)(6,106,14,116)(7,105,15,115)(8,104,16,114)(9,103,17,113)(10,102,18,112)(21,94,31,84)(22,93,32,83)(23,92,33,82)(24,91,34,81)(25,100,35,90)(26,99,36,89)(27,98,37,88)(28,97,38,87)(29,96,39,86)(30,95,40,85)(41,159,51,149)(42,158,52,148)(43,157,53,147)(44,156,54,146)(45,155,55,145)(46,154,56,144)(47,153,57,143)(48,152,58,142)(49,151,59,141)(50,160,60,150)(61,124,71,134)(62,123,72,133)(63,122,73,132)(64,121,74,131)(65,130,75,140)(66,129,76,139)(67,128,77,138)(68,127,78,137)(69,126,79,136)(70,125,80,135), (1,59,19,49)(2,60,20,50)(3,51,11,41)(4,52,12,42)(5,53,13,43)(6,54,14,44)(7,55,15,45)(8,56,16,46)(9,57,17,47)(10,58,18,48)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,131,91,121)(82,132,92,122)(83,133,93,123)(84,134,94,124)(85,135,95,125)(86,136,96,126)(87,137,97,127)(88,138,98,128)(89,139,99,129)(90,140,100,130)(101,151,111,141)(102,152,112,142)(103,153,113,143)(104,154,114,144)(105,155,115,145)(106,156,116,146)(107,157,117,147)(108,158,118,148)(109,159,119,149)(110,160,120,150) );
G=PermutationGroup([[(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)], [(1,96,6,91),(2,95,7,100),(3,94,8,99),(4,93,9,98),(5,92,10,97),(11,84,16,89),(12,83,17,88),(13,82,18,87),(14,81,19,86),(15,90,20,85),(21,114,26,119),(22,113,27,118),(23,112,28,117),(24,111,29,116),(25,120,30,115),(31,104,36,109),(32,103,37,108),(33,102,38,107),(34,101,39,106),(35,110,40,105),(41,134,46,139),(42,133,47,138),(43,132,48,137),(44,131,49,136),(45,140,50,135),(51,124,56,129),(52,123,57,128),(53,122,58,127),(54,121,59,126),(55,130,60,125),(61,154,66,159),(62,153,67,158),(63,152,68,157),(64,151,69,156),(65,160,70,155),(71,144,76,149),(72,143,77,148),(73,142,78,147),(74,141,79,146),(75,150,80,145)], [(1,101,19,111),(2,110,20,120),(3,109,11,119),(4,108,12,118),(5,107,13,117),(6,106,14,116),(7,105,15,115),(8,104,16,114),(9,103,17,113),(10,102,18,112),(21,94,31,84),(22,93,32,83),(23,92,33,82),(24,91,34,81),(25,100,35,90),(26,99,36,89),(27,98,37,88),(28,97,38,87),(29,96,39,86),(30,95,40,85),(41,159,51,149),(42,158,52,148),(43,157,53,147),(44,156,54,146),(45,155,55,145),(46,154,56,144),(47,153,57,143),(48,152,58,142),(49,151,59,141),(50,160,60,150),(61,124,71,134),(62,123,72,133),(63,122,73,132),(64,121,74,131),(65,130,75,140),(66,129,76,139),(67,128,77,138),(68,127,78,137),(69,126,79,136),(70,125,80,135)], [(1,59,19,49),(2,60,20,50),(3,51,11,41),(4,52,12,42),(5,53,13,43),(6,54,14,44),(7,55,15,45),(8,56,16,46),(9,57,17,47),(10,58,18,48),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70),(81,131,91,121),(82,132,92,122),(83,133,93,123),(84,134,94,124),(85,135,95,125),(86,136,96,126),(87,137,97,127),(88,138,98,128),(89,139,99,129),(90,140,100,130),(101,151,111,141),(102,152,112,142),(103,153,113,143),(104,154,114,144),(105,155,115,145),(106,156,116,146),(107,157,117,147),(108,158,118,148),(109,159,119,149),(110,160,120,150)]])
Dic5⋊3Q8 is a maximal subgroup of
Dic10⋊1C8 D4.D5⋊5C4 Dic5⋊6SD16 Dic10⋊2D4 Dic10.D4 C5⋊Q16⋊5C4 Dic5⋊4Q16 Dic5⋊Q16 Dic10.11D4 Dic5⋊8SD16 Dic20⋊15C4 Dic10⋊Q8 Dic10.Q8 Dic5⋊5Q16 Dic10⋊2Q8 Dic10.2Q8 C40⋊21(C2×C4) Dic10⋊C8 Dic5.M4(2) C20.M4(2) C10.82+ 1+4 C10.102+ 1+4 C10.52- 1+4 C42.87D10 C42.188D10 C42.94D10 C42.98D10 C4×D4⋊2D5 C42.106D10 C42.108D10 C42.114D10 Dic10⋊10Q8 C42.122D10 C4×Q8×D5 C42.125D10 Dic10⋊19D4 Dic10⋊20D4 C4⋊C4.178D10 C10.362+ 1+4 (Q8×Dic5)⋊C2 C10.152- 1+4 Dic10⋊21D4 Dic10⋊22D4 C10.522+ 1+4 C10.222- 1+4 C10.232- 1+4 C4⋊C4.197D10 C10.802- 1+4 C10.842- 1+4 C10.672+ 1+4 Dic10⋊7Q8 C42.236D10 C42.237D10 C42.151D10 C42.154D10 C42.155D10 C42.159D10 C42.160D10 C42.189D10 Dic10⋊8Q8 Dic10⋊9Q8 C42.241D10 C42.178D10 Dic5⋊5Dic6 Dic15⋊5Q8 Dic15⋊6Q8 Dic30⋊17C4 Dic15⋊10Q8
Dic5⋊3Q8 is a maximal quotient of
(C2×C20)⋊Q8 Dic5⋊2C42 C10.51(C4×D4) C2.(C4×D20) Dic5.5M4(2) Dic10⋊5C8 C42.198D10 C10.96(C4×D4) C20⋊4(C4⋊C4) (C2×Dic5)⋊6Q8 C4⋊C4×Dic5 C10.97(C4×D4) Dic5⋊5Dic6 Dic15⋊5Q8 Dic15⋊6Q8 Dic30⋊17C4 Dic15⋊10Q8
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | Q8 | D5 | C4○D4 | D10 | C4×D5 | D4⋊2D5 | Q8×D5 |
| kernel | Dic5⋊3Q8 | C4×Dic5 | C10.D4 | C5×C4⋊C4 | C2×Dic10 | Dic10 | Dic5 | C4⋊C4 | C10 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 8 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of Dic5⋊3Q8 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 9 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 40 | 0 | 0 |
| 35 | 35 | 0 | 0 |
| 0 | 0 | 1 | 21 |
| 0 | 0 | 37 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 1 |
| 0 | 0 | 17 | 33 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,1,0,0,0,0,1],[28,13,0,0,9,13,0,0,0,0,1,0,0,0,0,1],[6,35,0,0,40,35,0,0,0,0,1,37,0,0,21,40],[1,0,0,0,0,1,0,0,0,0,8,17,0,0,1,33] >;
Dic5⋊3Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_3Q_8 % in TeX
G:=Group("Dic5:3Q8"); // GroupNames label
G:=SmallGroup(160,108);
// by ID
G=gap.SmallGroup(160,108);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,116,122,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^4=1,b^2=a^5,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations