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## G = D10.5C42order 320 = 26·5

### 2nd non-split extension by D10 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10.5C42
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C42⋊D5 — D10.5C42
 Lower central C5 — C10 — D10.5C42
 Upper central C1 — C2×C8 — C4×C8

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, 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, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C82M4(2), C8×D5, C8⋊D5, C2×C52C8, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C42.D5, C8×Dic5, C408C4, C4×C40, C42⋊D5, D5×C2×C8, C2×C8⋊D5, D10.5C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, D10, C2×C42, C8○D4, C4×D5, C22×D5, C82M4(2), C2×C4×D5, D5×C42, D20.3C4, D10.5C42

Smallest permutation representation of D10.5C42
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)(81 146 106 126 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`

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