metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.7C42, C42.184D10, Dic5.7C42, (C8×D5)⋊10C4, C8⋊D5⋊7C4, C8.34(C4×D5), C8⋊C4⋊13D5, C40.78(C2×C4), C40⋊8C4⋊25C2, (C8×Dic5)⋊28C2, (C2×C8).271D10, C2.11(D5×C42), C10.46(C8○D4), C5⋊6(C8○2M4(2)), C10.30(C2×C42), D10⋊C4.17C4, C20.186(C22×C4), (C2×C20).814C23, (C2×C40).228C22, (C4×C20).229C22, C10.D4.17C4, C42⋊D5.11C2, C2.1(D20.2C4), (C4×Dic5).299C22, (C5×C8⋊C4)⋊9C2, (D5×C2×C8).27C2, C4.101(C2×C4×D5), (C4×C5⋊2C8)⋊21C2, (C2×C4).61(C4×D5), C22.41(C2×C4×D5), C5⋊2C8.28(C2×C4), (C4×D5).80(C2×C4), (C2×C20).321(C2×C4), (C2×C8⋊D5).16C2, (C2×C4×D5).341C22, (C2×Dic5).93(C2×C4), (C22×D5).71(C2×C4), (C2×C4).756(C22×D5), (C2×C10).170(C22×C4), (C2×C5⋊2C8).352C22, SmallGroup(320,335)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.7C42
G = < a,b,c,d | a10=b2=c4=1, d4=a5, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=a5c >
Subgroups: 350 in 130 conjugacy classes, 75 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4×C8, C8⋊C4, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C8○2M4(2), C8×D5, C8⋊D5, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C4×C5⋊2C8, C8×Dic5, C40⋊8C4, C5×C8⋊C4, C42⋊D5, D5×C2×C8, C2×C8⋊D5, D10.7C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, D10, C2×C42, C8○D4, C4×D5, C22×D5, C8○2M4(2), C2×C4×D5, D5×C42, D20.2C4, D10.7C42
►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 10)(2 9)(3 8)(4 7)(5 6)(11 14)(12 13)(15 20)(16 19)(17 18)(21 24)(22 23)(25 30)(26 29)(27 28)(31 34)(32 33)(35 40)(36 39)(37 38)(41 44)(42 43)(45 50)(46 49)(47 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 64)(62 63)(65 70)(66 69)(67 68)(71 74)(72 73)(75 80)(76 79)(77 78)(81 89)(82 88)(83 87)(84 86)(91 99)(92 98)(93 97)(94 96)(101 109)(102 108)(103 107)(104 106)(111 119)(112 118)(113 117)(114 116)(121 129)(122 128)(123 127)(124 126)(131 139)(132 138)(133 137)(134 136)(141 149)(142 148)(143 147)(144 146)(151 159)(152 158)(153 157)(154 156)
(1 93 13 83)(2 94 14 84)(3 95 15 85)(4 96 16 86)(5 97 17 87)(6 98 18 88)(7 99 19 89)(8 100 20 90)(9 91 11 81)(10 92 12 82)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(26 116 36 106)(27 117 37 107)(28 118 38 108)(29 119 39 109)(30 120 40 110)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(46 136 56 126)(47 137 57 127)(48 138 58 128)(49 139 59 129)(50 140 60 130)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)(66 156 76 146)(67 157 77 147)(68 158 78 148)(69 159 79 149)(70 160 80 150)
(1 68 28 48 6 63 23 43)(2 69 29 49 7 64 24 44)(3 70 30 50 8 65 25 45)(4 61 21 41 9 66 26 46)(5 62 22 42 10 67 27 47)(11 76 36 56 16 71 31 51)(12 77 37 57 17 72 32 52)(13 78 38 58 18 73 33 53)(14 79 39 59 19 74 34 54)(15 80 40 60 20 75 35 55)(81 141 106 121 86 146 101 126)(82 142 107 122 87 147 102 127)(83 143 108 123 88 148 103 128)(84 144 109 124 89 149 104 129)(85 145 110 125 90 150 105 130)(91 151 116 131 96 156 111 136)(92 152 117 132 97 157 112 137)(93 153 118 133 98 158 113 138)(94 154 119 134 99 159 114 139)(95 155 120 135 100 160 115 140)
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,34)(32,33)(35,40)(36,39)(37,38)(41,44)(42,43)(45,50)(46,49)(47,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,64)(62,63)(65,70)(66,69)(67,68)(71,74)(72,73)(75,80)(76,79)(77,78)(81,89)(82,88)(83,87)(84,86)(91,99)(92,98)(93,97)(94,96)(101,109)(102,108)(103,107)(104,106)(111,119)(112,118)(113,117)(114,116)(121,129)(122,128)(123,127)(124,126)(131,139)(132,138)(133,137)(134,136)(141,149)(142,148)(143,147)(144,146)(151,159)(152,158)(153,157)(154,156), (1,93,13,83)(2,94,14,84)(3,95,15,85)(4,96,16,86)(5,97,17,87)(6,98,18,88)(7,99,19,89)(8,100,20,90)(9,91,11,81)(10,92,12,82)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(26,116,36,106)(27,117,37,107)(28,118,38,108)(29,119,39,109)(30,120,40,110)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(46,136,56,126)(47,137,57,127)(48,138,58,128)(49,139,59,129)(50,140,60,130)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145)(66,156,76,146)(67,157,77,147)(68,158,78,148)(69,159,79,149)(70,160,80,150), (1,68,28,48,6,63,23,43)(2,69,29,49,7,64,24,44)(3,70,30,50,8,65,25,45)(4,61,21,41,9,66,26,46)(5,62,22,42,10,67,27,47)(11,76,36,56,16,71,31,51)(12,77,37,57,17,72,32,52)(13,78,38,58,18,73,33,53)(14,79,39,59,19,74,34,54)(15,80,40,60,20,75,35,55)(81,141,106,121,86,146,101,126)(82,142,107,122,87,147,102,127)(83,143,108,123,88,148,103,128)(84,144,109,124,89,149,104,129)(85,145,110,125,90,150,105,130)(91,151,116,131,96,156,111,136)(92,152,117,132,97,157,112,137)(93,153,118,133,98,158,113,138)(94,154,119,134,99,159,114,139)(95,155,120,135,100,160,115,140)>;
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,34)(32,33)(35,40)(36,39)(37,38)(41,44)(42,43)(45,50)(46,49)(47,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,64)(62,63)(65,70)(66,69)(67,68)(71,74)(72,73)(75,80)(76,79)(77,78)(81,89)(82,88)(83,87)(84,86)(91,99)(92,98)(93,97)(94,96)(101,109)(102,108)(103,107)(104,106)(111,119)(112,118)(113,117)(114,116)(121,129)(122,128)(123,127)(124,126)(131,139)(132,138)(133,137)(134,136)(141,149)(142,148)(143,147)(144,146)(151,159)(152,158)(153,157)(154,156), (1,93,13,83)(2,94,14,84)(3,95,15,85)(4,96,16,86)(5,97,17,87)(6,98,18,88)(7,99,19,89)(8,100,20,90)(9,91,11,81)(10,92,12,82)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(26,116,36,106)(27,117,37,107)(28,118,38,108)(29,119,39,109)(30,120,40,110)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(46,136,56,126)(47,137,57,127)(48,138,58,128)(49,139,59,129)(50,140,60,130)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145)(66,156,76,146)(67,157,77,147)(68,158,78,148)(69,159,79,149)(70,160,80,150), (1,68,28,48,6,63,23,43)(2,69,29,49,7,64,24,44)(3,70,30,50,8,65,25,45)(4,61,21,41,9,66,26,46)(5,62,22,42,10,67,27,47)(11,76,36,56,16,71,31,51)(12,77,37,57,17,72,32,52)(13,78,38,58,18,73,33,53)(14,79,39,59,19,74,34,54)(15,80,40,60,20,75,35,55)(81,141,106,121,86,146,101,126)(82,142,107,122,87,147,102,127)(83,143,108,123,88,148,103,128)(84,144,109,124,89,149,104,129)(85,145,110,125,90,150,105,130)(91,151,116,131,96,156,111,136)(92,152,117,132,97,157,112,137)(93,153,118,133,98,158,113,138)(94,154,119,134,99,159,114,139)(95,155,120,135,100,160,115,140) );
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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13),(15,20),(16,19),(17,18),(21,24),(22,23),(25,30),(26,29),(27,28),(31,34),(32,33),(35,40),(36,39),(37,38),(41,44),(42,43),(45,50),(46,49),(47,48),(51,54),(52,53),(55,60),(56,59),(57,58),(61,64),(62,63),(65,70),(66,69),(67,68),(71,74),(72,73),(75,80),(76,79),(77,78),(81,89),(82,88),(83,87),(84,86),(91,99),(92,98),(93,97),(94,96),(101,109),(102,108),(103,107),(104,106),(111,119),(112,118),(113,117),(114,116),(121,129),(122,128),(123,127),(124,126),(131,139),(132,138),(133,137),(134,136),(141,149),(142,148),(143,147),(144,146),(151,159),(152,158),(153,157),(154,156)], [(1,93,13,83),(2,94,14,84),(3,95,15,85),(4,96,16,86),(5,97,17,87),(6,98,18,88),(7,99,19,89),(8,100,20,90),(9,91,11,81),(10,92,12,82),(21,111,31,101),(22,112,32,102),(23,113,33,103),(24,114,34,104),(25,115,35,105),(26,116,36,106),(27,117,37,107),(28,118,38,108),(29,119,39,109),(30,120,40,110),(41,131,51,121),(42,132,52,122),(43,133,53,123),(44,134,54,124),(45,135,55,125),(46,136,56,126),(47,137,57,127),(48,138,58,128),(49,139,59,129),(50,140,60,130),(61,151,71,141),(62,152,72,142),(63,153,73,143),(64,154,74,144),(65,155,75,145),(66,156,76,146),(67,157,77,147),(68,158,78,148),(69,159,79,149),(70,160,80,150)], [(1,68,28,48,6,63,23,43),(2,69,29,49,7,64,24,44),(3,70,30,50,8,65,25,45),(4,61,21,41,9,66,26,46),(5,62,22,42,10,67,27,47),(11,76,36,56,16,71,31,51),(12,77,37,57,17,72,32,52),(13,78,38,58,18,73,33,53),(14,79,39,59,19,74,34,54),(15,80,40,60,20,75,35,55),(81,141,106,121,86,146,101,126),(82,142,107,122,87,147,102,127),(83,143,108,123,88,148,103,128),(84,144,109,124,89,149,104,129),(85,145,110,125,90,150,105,130),(91,151,116,131,96,156,111,136),(92,152,117,132,97,157,112,137),(93,153,118,133,98,158,113,138),(94,154,119,134,99,159,114,139),(95,155,120,135,100,160,115,140)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.2C4 |
| kernel | D10.7C42 | C4×C5⋊2C8 | C8×Dic5 | C40⋊8C4 | C5×C8⋊C4 | C42⋊D5 | D5×C2×C8 | C2×C8⋊D5 | C8×D5 | C8⋊D5 | C10.D4 | D10⋊C4 | C8⋊C4 | C42 | C2×C8 | C10 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 4 | 4 | 2 | 2 | 4 | 8 | 16 | 8 | 8 |
Matrix representation of D10.7C42 ►in GL4(𝔽41) generated by
| 40 | 35 | 0 | 0 |
| 6 | 35 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 6 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 4 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 15 | 14 |
G:=sub<GL(4,GF(41))| [40,6,0,0,35,35,0,0,0,0,40,0,0,0,0,40],[40,6,0,0,0,1,0,0,0,0,40,4,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,21,40],[40,0,0,0,0,40,0,0,0,0,27,15,0,0,0,14] >;
D10.7C42 in GAP, Magma, Sage, TeX
D_{10}._7C_4^2 % in TeX
G:=Group("D10.7C4^2"); // GroupNames label
G:=SmallGroup(320,335);
// by ID
G=gap.SmallGroup(320,335);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^4=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*c>;
// generators/relations