metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.5C42, C42.242D10, Dic5.5C42, (C8×D5)⋊5C4, (C4×C8)⋊15D5, (C4×C40)⋊22C2, C8⋊D5⋊8C4, C8.42(C4×D5), C40⋊8C4⋊31C2, C2.8(D5×C42), C40.100(C2×C4), (C8×Dic5)⋊12C2, (C2×C8).340D10, C10.29(C8○D4), C5⋊5(C8○2M4(2)), C10.27(C2×C42), D10⋊C4.29C4, C20.183(C22×C4), (C2×C20).807C23, (C4×C20).340C22, (C2×C40).406C22, C10.D4.29C4, C42.D5⋊24C2, C42⋊D5.14C2, C2.3(D20.3C4), (C4×Dic5).296C22, C4.98(C2×C4×D5), (D5×C2×C8).14C2, (C2×C4).88(C4×D5), C22.38(C2×C4×D5), C5⋊2C8.27(C2×C4), (C4×D5).79(C2×C4), (C2×C20).378(C2×C4), (C2×C8⋊D5).18C2, (C2×C4×D5).338C22, (C2×Dic5).92(C2×C4), (C22×D5).69(C2×C4), (C2×C4).749(C22×D5), (C2×C10).163(C22×C4), (C2×C5⋊2C8).300C22, SmallGroup(320,316)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.5C42
G = < a,b,c,d | a10=b2=c4=1, d4=a5, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, cd=dc >
Subgroups: 350 in 130 conjugacy classes, 75 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, D5 [×2], C10, C10 [×2], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4×C8, C4×C8, C8⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C5⋊2C8 [×4], C40 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C8○2M4(2), C8×D5 [×4], C8⋊D5 [×4], C2×C5⋊2C8 [×2], C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C42.D5, C8×Dic5, C40⋊8C4, C4×C40, C42⋊D5, D5×C2×C8, C2×C8⋊D5, D10.5C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], C22×C4 [×3], D10 [×3], C2×C42, C8○D4 [×2], C4×D5 [×6], C22×D5, C8○2M4(2), C2×C4×D5 [×3], D5×C42, D20.3C4 [×2], D10.5C42
(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)(81 146 106 126 86 141 101 121)(82 147 107 127 87 142 102 122)(83 148 108 128 88 143 103 123)(84 149 109 129 89 144 104 124)(85 150 110 130 90 145 105 125)(91 156 116 136 96 151 111 131)(92 157 117 137 97 152 112 132)(93 158 118 138 98 153 113 133)(94 159 119 139 99 154 114 134)(95 160 120 140 100 155 115 135)
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)(81,146,106,126,86,141,101,121)(82,147,107,127,87,142,102,122)(83,148,108,128,88,143,103,123)(84,149,109,129,89,144,104,124)(85,150,110,130,90,145,105,125)(91,156,116,136,96,151,111,131)(92,157,117,137,97,152,112,132)(93,158,118,138,98,153,113,133)(94,159,119,139,99,154,114,134)(95,160,120,140,100,155,115,135)>;
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)(81,146,106,126,86,141,101,121)(82,147,107,127,87,142,102,122)(83,148,108,128,88,143,103,123)(84,149,109,129,89,144,104,124)(85,150,110,130,90,145,105,125)(91,156,116,136,96,151,111,131)(92,157,117,137,97,152,112,132)(93,158,118,138,98,153,113,133)(94,159,119,139,99,154,114,134)(95,160,120,140,100,155,115,135) );
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),(81,146,106,126,86,141,101,121),(82,147,107,127,87,142,102,122),(83,148,108,128,88,143,103,123),(84,149,109,129,89,144,104,124),(85,150,110,130,90,145,105,125),(91,156,116,136,96,151,111,131),(92,157,117,137,97,152,112,132),(93,158,118,138,98,153,113,133),(94,159,119,139,99,154,114,134),(95,160,120,140,100,155,115,135)])
104 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
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 | 40 | ··· | 40 |
size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 |
kernel | D10.5C42 | C42.D5 | C8×Dic5 | C40⋊8C4 | C4×C40 | C42⋊D5 | D5×C2×C8 | C2×C8⋊D5 | C8×D5 | C8⋊D5 | C10.D4 | D10⋊C4 | C4×C8 | 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 | 32 |
Matrix representation of D10.5C42 ►in GL5(𝔽41)
1 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 7 |
0 | 0 | 0 | 35 | 6 |
40 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 |
0 | 32 | 1 | 0 | 0 |
0 | 0 | 0 | 35 | 7 |
0 | 0 | 0 | 36 | 6 |
9 | 0 | 0 | 0 | 0 |
0 | 9 | 39 | 0 | 0 |
0 | 40 | 32 | 0 | 0 |
0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 40 |
40 | 0 | 0 | 0 | 0 |
0 | 3 | 0 | 0 | 0 |
0 | 0 | 3 | 0 | 0 |
0 | 0 | 0 | 32 | 0 |
0 | 0 | 0 | 0 | 32 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,35,0,0,0,7,6],[40,0,0,0,0,0,40,32,0,0,0,0,1,0,0,0,0,0,35,36,0,0,0,7,6],[9,0,0,0,0,0,9,40,0,0,0,39,32,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,32,0,0,0,0,0,32] >;
D10.5C42 in GAP, Magma, Sage, TeX
D_{10}._5C_4^2
% in TeX
G:=Group("D10.5C4^2");
// GroupNames label
G:=SmallGroup(320,316);
// by ID
G=gap.SmallGroup(320,316);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,58,136,12550]);
// 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,c*d=d*c>;
// generators/relations