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## G = D10.6C42order 320 = 26·5

### 3rd non-split extension by D10 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10.6C42
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C42 — D10.6C42
 Lower central C5 — C10 — D10.6C42
 Upper central C1 — C2×C4 — C8⋊C4

Generators and relations for D10.6C42
G = < a,b,c,d | a10=b2=c4=1, d4=a5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=a5c >

Subgroups: 398 in 142 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, C20, D10, D10, C2×C10, C4×C8, C8⋊C4, C8⋊C4, C2×C42, C2×M4(2), C52C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×M4(2), C8⋊D5, C2×C52C8, C4×Dic5, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C2×C4×D5, C42.D5, C8×Dic5, C5×C8⋊C4, D5×C42, C2×C8⋊D5, D10.6C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, M4(2), C22×C4, D10, C2×C42, C2×M4(2), C4×D5, C22×D5, C4×M4(2), C2×C4×D5, D5×C42, D5×M4(2), D10.6C42

Smallest permutation representation of D10.6C42
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 49)(42 48)(43 47)(44 46)(51 59)(52 58)(53 57)(54 56)(61 69)(62 68)(63 67)(64 66)(71 79)(72 78)(73 77)(74 76)(81 84)(82 83)(85 90)(86 89)(87 88)(91 94)(92 93)(95 100)(96 99)(97 98)(101 104)(102 103)(105 110)(106 109)(107 108)(111 114)(112 113)(115 120)(116 119)(117 118)(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)(41,49)(42,48)(43,47)(44,46)(51,59)(52,58)(53,57)(54,56)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(81,84)(82,83)(85,90)(86,89)(87,88)(91,94)(92,93)(95,100)(96,99)(97,98)(101,104)(102,103)(105,110)(106,109)(107,108)(111,114)(112,113)(115,120)(116,119)(117,118)(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)(41,49)(42,48)(43,47)(44,46)(51,59)(52,58)(53,57)(54,56)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)(81,84)(82,83)(85,90)(86,89)(87,88)(91,94)(92,93)(95,100)(96,99)(97,98)(101,104)(102,103)(105,110)(106,109)(107,108)(111,114)(112,113)(115,120)(116,119)(117,118)(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),(41,49),(42,48),(43,47),(44,46),(51,59),(52,58),(53,57),(54,56),(61,69),(62,68),(63,67),(64,66),(71,79),(72,78),(73,77),(74,76),(81,84),(82,83),(85,90),(86,89),(87,88),(91,94),(92,93),(95,100),(96,99),(97,98),(101,104),(102,103),(105,110),(106,109),(107,108),(111,114),(112,113),(115,120),(116,119),(117,118),(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 ··· 4P 4Q 4R 5A 5B 8A ··· 8H 8I ··· 8P 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 4 4 5 5 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 5 ··· 5 10 10 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D5 M4(2) D10 D10 C4×D5 C4×D5 D5×M4(2) kernel D10.6C42 C42.D5 C8×Dic5 C5×C8⋊C4 D5×C42 C2×C8⋊D5 C8⋊D5 C4×Dic5 C2×C4×D5 C8⋊C4 Dic5 C42 C2×C8 C8 C2×C4 C2 # reps 1 1 2 1 1 2 16 4 4 2 8 2 4 16 8 8

Matrix representation of D10.6C42 in GL5(𝔽41)

 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 35 0 0 0 6 35
,
 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 35 40 0 0 0 35 6
,
 9 0 0 0 0 0 32 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 1
,
 32 0 0 0 0 0 0 1 0 0 0 9 0 0 0 0 0 0 32 0 0 0 0 0 32

`G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,6,0,0,0,35,35],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,35,35,0,0,0,40,6],[9,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,0,9,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,32] >;`

D10.6C42 in GAP, Magma, Sage, TeX

`D_{10}._6C_4^2`
`% in TeX`

`G:=Group("D10.6C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,334);`
`// by ID`

`G=gap.SmallGroup(320,334);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,387,58,136,12550]);`
`// 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,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^5*c>;`
`// generators/relations`

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