metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊5M4(2), C20.50(C4⋊C4), C20.87(C2×Q8), (C2×C20).24Q8, (C2×C20).164D4, C20.438(C2×D4), (C2×C8).185D10, C23.53(C4×D5), C5⋊6(C4⋊M4(2)), C20.8Q8⋊38C2, (C4×Dic5).12C4, (C2×C4).34Dic10, C4.52(C2×Dic10), C2.19(D5×M4(2)), (C2×C20).863C23, (C2×C40).315C22, (C22×C4).345D10, (C2×M4(2)).12D5, C10.64(C2×M4(2)), C4.18(C10.D4), (C10×M4(2)).23C2, (C22×Dic5).21C4, (C22×C20).177C22, (C4×Dic5).315C22, C22.10(C10.D4), C10.71(C2×C4⋊C4), (C2×C4).157(C4×D5), C4.128(C2×C5⋊D4), (C2×C4×Dic5).10C2, (C2×C10).42(C4⋊C4), C22.143(C2×C4×D5), (C2×C20).270(C2×C4), (C2×C4).140(C5⋊D4), C2.16(C2×C10.D4), (C2×C4).805(C22×D5), (C2×C4.Dic5).22C2, (C22×C10).131(C2×C4), (C2×C10).234(C22×C4), (C2×C5⋊2C8).210C22, (C2×Dic5).156(C2×C4), SmallGroup(320,745)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊5M4(2)
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c5 >
Subgroups: 334 in 126 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C2×C42, C2×M4(2), C2×M4(2), C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4⋊M4(2), C2×C5⋊2C8, C4.Dic5, C4×Dic5, C4×Dic5, C2×C40, C5×M4(2), C22×Dic5, C22×C20, C20.8Q8, C2×C4.Dic5, C2×C4×Dic5, C10×M4(2), Dic5⋊5M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C2×M4(2), Dic10, C4×D5, C5⋊D4, C22×D5, C4⋊M4(2), C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, D5×M4(2), C2×C10.D4, Dic5⋊5M4(2)
►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)
(1 111 6 116)(2 120 7 115)(3 119 8 114)(4 118 9 113)(5 117 10 112)(11 63 16 68)(12 62 17 67)(13 61 18 66)(14 70 19 65)(15 69 20 64)(21 121 26 126)(22 130 27 125)(23 129 28 124)(24 128 29 123)(25 127 30 122)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 109 46 104)(42 108 47 103)(43 107 48 102)(44 106 49 101)(45 105 50 110)(51 154 56 159)(52 153 57 158)(53 152 58 157)(54 151 59 156)(55 160 60 155)(71 150 76 145)(72 149 77 144)(73 148 78 143)(74 147 79 142)(75 146 80 141)(81 140 86 135)(82 139 87 134)(83 138 88 133)(84 137 89 132)(85 136 90 131)
(1 90 43 54 25 74 34 63)(2 81 44 55 26 75 35 64)(3 82 45 56 27 76 36 65)(4 83 46 57 28 77 37 66)(5 84 47 58 29 78 38 67)(6 85 48 59 30 79 39 68)(7 86 49 60 21 80 40 69)(8 87 50 51 22 71 31 70)(9 88 41 52 23 72 32 61)(10 89 42 53 24 73 33 62)(11 111 136 107 156 127 142 91)(12 112 137 108 157 128 143 92)(13 113 138 109 158 129 144 93)(14 114 139 110 159 130 145 94)(15 115 140 101 160 121 146 95)(16 116 131 102 151 122 147 96)(17 117 132 103 152 123 148 97)(18 118 133 104 153 124 149 98)(19 119 134 105 154 125 150 99)(20 120 135 106 155 126 141 100)
(11 156)(12 157)(13 158)(14 159)(15 160)(16 151)(17 152)(18 153)(19 154)(20 155)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 87)(72 88)(73 89)(74 90)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
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), (1,111,6,116)(2,120,7,115)(3,119,8,114)(4,118,9,113)(5,117,10,112)(11,63,16,68)(12,62,17,67)(13,61,18,66)(14,70,19,65)(15,69,20,64)(21,121,26,126)(22,130,27,125)(23,129,28,124)(24,128,29,123)(25,127,30,122)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,109,46,104)(42,108,47,103)(43,107,48,102)(44,106,49,101)(45,105,50,110)(51,154,56,159)(52,153,57,158)(53,152,58,157)(54,151,59,156)(55,160,60,155)(71,150,76,145)(72,149,77,144)(73,148,78,143)(74,147,79,142)(75,146,80,141)(81,140,86,135)(82,139,87,134)(83,138,88,133)(84,137,89,132)(85,136,90,131), (1,90,43,54,25,74,34,63)(2,81,44,55,26,75,35,64)(3,82,45,56,27,76,36,65)(4,83,46,57,28,77,37,66)(5,84,47,58,29,78,38,67)(6,85,48,59,30,79,39,68)(7,86,49,60,21,80,40,69)(8,87,50,51,22,71,31,70)(9,88,41,52,23,72,32,61)(10,89,42,53,24,73,33,62)(11,111,136,107,156,127,142,91)(12,112,137,108,157,128,143,92)(13,113,138,109,158,129,144,93)(14,114,139,110,159,130,145,94)(15,115,140,101,160,121,146,95)(16,116,131,102,151,122,147,96)(17,117,132,103,152,123,148,97)(18,118,133,104,153,124,149,98)(19,119,134,105,154,125,150,99)(20,120,135,106,155,126,141,100), (11,156)(12,157)(13,158)(14,159)(15,160)(16,151)(17,152)(18,153)(19,154)(20,155)(51,70)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)>;
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), (1,111,6,116)(2,120,7,115)(3,119,8,114)(4,118,9,113)(5,117,10,112)(11,63,16,68)(12,62,17,67)(13,61,18,66)(14,70,19,65)(15,69,20,64)(21,121,26,126)(22,130,27,125)(23,129,28,124)(24,128,29,123)(25,127,30,122)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,109,46,104)(42,108,47,103)(43,107,48,102)(44,106,49,101)(45,105,50,110)(51,154,56,159)(52,153,57,158)(53,152,58,157)(54,151,59,156)(55,160,60,155)(71,150,76,145)(72,149,77,144)(73,148,78,143)(74,147,79,142)(75,146,80,141)(81,140,86,135)(82,139,87,134)(83,138,88,133)(84,137,89,132)(85,136,90,131), (1,90,43,54,25,74,34,63)(2,81,44,55,26,75,35,64)(3,82,45,56,27,76,36,65)(4,83,46,57,28,77,37,66)(5,84,47,58,29,78,38,67)(6,85,48,59,30,79,39,68)(7,86,49,60,21,80,40,69)(8,87,50,51,22,71,31,70)(9,88,41,52,23,72,32,61)(10,89,42,53,24,73,33,62)(11,111,136,107,156,127,142,91)(12,112,137,108,157,128,143,92)(13,113,138,109,158,129,144,93)(14,114,139,110,159,130,145,94)(15,115,140,101,160,121,146,95)(16,116,131,102,151,122,147,96)(17,117,132,103,152,123,148,97)(18,118,133,104,153,124,149,98)(19,119,134,105,154,125,150,99)(20,120,135,106,155,126,141,100), (11,156)(12,157)(13,158)(14,159)(15,160)(16,151)(17,152)(18,153)(19,154)(20,155)(51,70)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(71,87)(72,88)(73,89)(74,90)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146) );
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)], [(1,111,6,116),(2,120,7,115),(3,119,8,114),(4,118,9,113),(5,117,10,112),(11,63,16,68),(12,62,17,67),(13,61,18,66),(14,70,19,65),(15,69,20,64),(21,121,26,126),(22,130,27,125),(23,129,28,124),(24,128,29,123),(25,127,30,122),(31,94,36,99),(32,93,37,98),(33,92,38,97),(34,91,39,96),(35,100,40,95),(41,109,46,104),(42,108,47,103),(43,107,48,102),(44,106,49,101),(45,105,50,110),(51,154,56,159),(52,153,57,158),(53,152,58,157),(54,151,59,156),(55,160,60,155),(71,150,76,145),(72,149,77,144),(73,148,78,143),(74,147,79,142),(75,146,80,141),(81,140,86,135),(82,139,87,134),(83,138,88,133),(84,137,89,132),(85,136,90,131)], [(1,90,43,54,25,74,34,63),(2,81,44,55,26,75,35,64),(3,82,45,56,27,76,36,65),(4,83,46,57,28,77,37,66),(5,84,47,58,29,78,38,67),(6,85,48,59,30,79,39,68),(7,86,49,60,21,80,40,69),(8,87,50,51,22,71,31,70),(9,88,41,52,23,72,32,61),(10,89,42,53,24,73,33,62),(11,111,136,107,156,127,142,91),(12,112,137,108,157,128,143,92),(13,113,138,109,158,129,144,93),(14,114,139,110,159,130,145,94),(15,115,140,101,160,121,146,95),(16,116,131,102,151,122,147,96),(17,117,132,103,152,123,148,97),(18,118,133,104,153,124,149,98),(19,119,134,105,154,125,150,99),(20,120,135,106,155,126,141,100)], [(11,156),(12,157),(13,158),(14,159),(15,160),(16,151),(17,152),(18,153),(19,154),(20,155),(51,70),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(71,87),(72,88),(73,89),(74,90),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | M4(2) | D10 | D10 | Dic10 | C4×D5 | C5⋊D4 | C4×D5 | D5×M4(2) |
| kernel | Dic5⋊5M4(2) | C20.8Q8 | C2×C4.Dic5 | C2×C4×Dic5 | C10×M4(2) | C4×Dic5 | C22×Dic5 | C2×C20 | C2×C20 | C2×M4(2) | Dic5 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 8 | 4 | 2 | 8 | 4 | 8 | 4 | 8 |
Matrix representation of Dic5⋊5M4(2) ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 7 | 40 |
| 0 | 0 | 1 | 0 |
| 32 | 16 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 27 | 28 |
| 0 | 0 | 12 | 14 |
| 9 | 2 | 0 | 0 |
| 5 | 32 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 40 | 24 |
| 1 | 21 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,7,1,0,0,40,0],[32,0,0,0,16,9,0,0,0,0,27,12,0,0,28,14],[9,5,0,0,2,32,0,0,0,0,17,40,0,0,1,24],[1,0,0,0,21,40,0,0,0,0,1,0,0,0,0,1] >;
Dic5⋊5M4(2) in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_5M_4(2) % in TeX
G:=Group("Dic5:5M4(2)"); // GroupNames label
G:=SmallGroup(320,745);
// by ID
G=gap.SmallGroup(320,745);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^5>;
// generators/relations