metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.9M4(2), C40⋊8C4⋊11C2, C22⋊C8.7D5, (C2×C8).162D10, C23.45(C4×D5), C5⋊6(C42.6C4), C20.8Q8⋊16C2, (C4×Dic5).21C4, C2.11(D5×M4(2)), C20.295(C4○D4), (C2×C20).817C23, (C2×C40).168C22, (C22×C4).301D10, C22.7(C8⋊D5), C10.54(C2×M4(2)), (C2×C10).16M4(2), C4.121(D4⋊2D5), C20.55D4.14C2, (C22×Dic5).14C4, C10.40(C42⋊C2), (C22×C20).335C22, (C4×Dic5).300C22, C2.9(C23.11D10), C2.8(C2×C8⋊D5), (C2×C4).130(C4×D5), (C2×C4×Dic5).31C2, C22.101(C2×C4×D5), (C2×C20).324(C2×C4), (C5×C22⋊C8).10C2, (C2×C4).759(C22×D5), (C2×C10).173(C22×C4), (C22×C10).103(C2×C4), (C2×C5⋊2C8).194C22, (C2×Dic5).138(C2×C4), SmallGroup(320,346)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.9M4(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=a5c5 >
Subgroups: 302 in 110 conjugacy classes, 53 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.6C4, C2×C5⋊2C8, C4×Dic5, C2×C40, C22×Dic5, C22×C20, C20.8Q8, C40⋊8C4, C20.55D4, C5×C22⋊C8, C2×C4×Dic5, Dic5.9M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, C4○D4, D10, C42⋊C2, C2×M4(2), C4×D5, C22×D5, C42.6C4, C8⋊D5, C2×C4×D5, D4⋊2D5, C23.11D10, C2×C8⋊D5, D5×M4(2), Dic5.9M4(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)
(1 127 6 122)(2 126 7 121)(3 125 8 130)(4 124 9 129)(5 123 10 128)(11 59 16 54)(12 58 17 53)(13 57 18 52)(14 56 19 51)(15 55 20 60)(21 111 26 116)(22 120 27 115)(23 119 28 114)(24 118 29 113)(25 117 30 112)(31 110 36 105)(32 109 37 104)(33 108 38 103)(34 107 39 102)(35 106 40 101)(41 93 46 98)(42 92 47 97)(43 91 48 96)(44 100 49 95)(45 99 50 94)(61 158 66 153)(62 157 67 152)(63 156 68 151)(64 155 69 160)(65 154 70 159)(71 139 76 134)(72 138 77 133)(73 137 78 132)(74 136 79 131)(75 135 80 140)(81 141 86 146)(82 150 87 145)(83 149 88 144)(84 148 89 143)(85 147 90 142)
(1 85 48 54 21 79 39 63)(2 86 49 55 22 80 40 64)(3 87 50 56 23 71 31 65)(4 88 41 57 24 72 32 66)(5 89 42 58 25 73 33 67)(6 90 43 59 26 74 34 68)(7 81 44 60 27 75 35 69)(8 82 45 51 28 76 36 70)(9 83 46 52 29 77 37 61)(10 84 47 53 30 78 38 62)(11 116 131 107 156 122 147 91)(12 117 132 108 157 123 148 92)(13 118 133 109 158 124 149 93)(14 119 134 110 159 125 150 94)(15 120 135 101 160 126 141 95)(16 111 136 102 151 127 142 96)(17 112 137 103 152 128 143 97)(18 113 138 104 153 129 144 98)(19 114 139 105 154 130 145 99)(20 115 140 106 155 121 146 100)
(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 61)(58 62)(59 63)(60 64)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 81)(131 142)(132 143)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 141)
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), (1,127,6,122)(2,126,7,121)(3,125,8,130)(4,124,9,129)(5,123,10,128)(11,59,16,54)(12,58,17,53)(13,57,18,52)(14,56,19,51)(15,55,20,60)(21,111,26,116)(22,120,27,115)(23,119,28,114)(24,118,29,113)(25,117,30,112)(31,110,36,105)(32,109,37,104)(33,108,38,103)(34,107,39,102)(35,106,40,101)(41,93,46,98)(42,92,47,97)(43,91,48,96)(44,100,49,95)(45,99,50,94)(61,158,66,153)(62,157,67,152)(63,156,68,151)(64,155,69,160)(65,154,70,159)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,141,86,146)(82,150,87,145)(83,149,88,144)(84,148,89,143)(85,147,90,142), (1,85,48,54,21,79,39,63)(2,86,49,55,22,80,40,64)(3,87,50,56,23,71,31,65)(4,88,41,57,24,72,32,66)(5,89,42,58,25,73,33,67)(6,90,43,59,26,74,34,68)(7,81,44,60,27,75,35,69)(8,82,45,51,28,76,36,70)(9,83,46,52,29,77,37,61)(10,84,47,53,30,78,38,62)(11,116,131,107,156,122,147,91)(12,117,132,108,157,123,148,92)(13,118,133,109,158,124,149,93)(14,119,134,110,159,125,150,94)(15,120,135,101,160,126,141,95)(16,111,136,102,151,127,142,96)(17,112,137,103,152,128,143,97)(18,113,138,104,153,129,144,98)(19,114,139,105,154,130,145,99)(20,115,140,106,155,121,146,100), (11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,61)(58,62)(59,63)(60,64)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,81)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141)>;
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), (1,127,6,122)(2,126,7,121)(3,125,8,130)(4,124,9,129)(5,123,10,128)(11,59,16,54)(12,58,17,53)(13,57,18,52)(14,56,19,51)(15,55,20,60)(21,111,26,116)(22,120,27,115)(23,119,28,114)(24,118,29,113)(25,117,30,112)(31,110,36,105)(32,109,37,104)(33,108,38,103)(34,107,39,102)(35,106,40,101)(41,93,46,98)(42,92,47,97)(43,91,48,96)(44,100,49,95)(45,99,50,94)(61,158,66,153)(62,157,67,152)(63,156,68,151)(64,155,69,160)(65,154,70,159)(71,139,76,134)(72,138,77,133)(73,137,78,132)(74,136,79,131)(75,135,80,140)(81,141,86,146)(82,150,87,145)(83,149,88,144)(84,148,89,143)(85,147,90,142), (1,85,48,54,21,79,39,63)(2,86,49,55,22,80,40,64)(3,87,50,56,23,71,31,65)(4,88,41,57,24,72,32,66)(5,89,42,58,25,73,33,67)(6,90,43,59,26,74,34,68)(7,81,44,60,27,75,35,69)(8,82,45,51,28,76,36,70)(9,83,46,52,29,77,37,61)(10,84,47,53,30,78,38,62)(11,116,131,107,156,122,147,91)(12,117,132,108,157,123,148,92)(13,118,133,109,158,124,149,93)(14,119,134,110,159,125,150,94)(15,120,135,101,160,126,141,95)(16,111,136,102,151,127,142,96)(17,112,137,103,152,128,143,97)(18,113,138,104,153,129,144,98)(19,114,139,105,154,130,145,99)(20,115,140,106,155,121,146,100), (11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,61)(58,62)(59,63)(60,64)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,81)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141) );
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)], [(1,127,6,122),(2,126,7,121),(3,125,8,130),(4,124,9,129),(5,123,10,128),(11,59,16,54),(12,58,17,53),(13,57,18,52),(14,56,19,51),(15,55,20,60),(21,111,26,116),(22,120,27,115),(23,119,28,114),(24,118,29,113),(25,117,30,112),(31,110,36,105),(32,109,37,104),(33,108,38,103),(34,107,39,102),(35,106,40,101),(41,93,46,98),(42,92,47,97),(43,91,48,96),(44,100,49,95),(45,99,50,94),(61,158,66,153),(62,157,67,152),(63,156,68,151),(64,155,69,160),(65,154,70,159),(71,139,76,134),(72,138,77,133),(73,137,78,132),(74,136,79,131),(75,135,80,140),(81,141,86,146),(82,150,87,145),(83,149,88,144),(84,148,89,143),(85,147,90,142)], [(1,85,48,54,21,79,39,63),(2,86,49,55,22,80,40,64),(3,87,50,56,23,71,31,65),(4,88,41,57,24,72,32,66),(5,89,42,58,25,73,33,67),(6,90,43,59,26,74,34,68),(7,81,44,60,27,75,35,69),(8,82,45,51,28,76,36,70),(9,83,46,52,29,77,37,61),(10,84,47,53,30,78,38,62),(11,116,131,107,156,122,147,91),(12,117,132,108,157,123,148,92),(13,118,133,109,158,124,149,93),(14,119,134,110,159,125,150,94),(15,120,135,101,160,126,141,95),(16,111,136,102,151,127,142,96),(17,112,137,103,152,128,143,97),(18,113,138,104,153,129,144,98),(19,114,139,105,154,130,145,99),(20,115,140,106,155,121,146,100)], [(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(57,61),(58,62),(59,63),(60,64),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,81),(131,142),(132,143),(133,144),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,141)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D5 | M4(2) | C4○D4 | M4(2) | D10 | D10 | C4×D5 | C4×D5 | C8⋊D5 | D4⋊2D5 | D5×M4(2) |
| kernel | Dic5.9M4(2) | C20.8Q8 | C40⋊8C4 | C20.55D4 | C5×C22⋊C8 | C2×C4×Dic5 | C4×Dic5 | C22×Dic5 | C22⋊C8 | Dic5 | C20 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 4 | 4 | 4 | 2 | 4 | 4 | 16 | 4 | 4 |
Matrix representation of Dic5.9M4(2) ►in GL4(𝔽41) generated by
| 34 | 40 | 0 | 0 |
| 8 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 13 | 39 | 0 | 0 |
| 3 | 28 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 18 | 32 |
| 33 | 39 | 0 | 0 |
| 16 | 8 | 0 | 0 |
| 0 | 0 | 4 | 37 |
| 0 | 0 | 14 | 37 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 40 |
G:=sub<GL(4,GF(41))| [34,8,0,0,40,1,0,0,0,0,40,0,0,0,0,40],[13,3,0,0,39,28,0,0,0,0,9,18,0,0,0,32],[33,16,0,0,39,8,0,0,0,0,4,14,0,0,37,37],[1,0,0,0,0,1,0,0,0,0,1,2,0,0,0,40] >;
Dic5.9M4(2) in GAP, Magma, Sage, TeX
{\rm Dic}_5._9M_4(2) % in TeX
G:=Group("Dic5.9M4(2)"); // GroupNames label
G:=SmallGroup(320,346);
// by ID
G=gap.SmallGroup(320,346);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,219,58,136,12550]);
// 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=a^5*c^5>;
// generators/relations