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## G = C42.164D10order 320 = 26·5

### 164th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.164D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10⋊2Q8 — C42.164D10
 Lower central C5 — C2×C10 — C42.164D10
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C42.164D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 798 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×13], C22, C22 [×9], C5, C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], D5 [×2], C10 [×3], C10, C42, C42 [×3], C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×3], C2×D4 [×3], C2×Q8 [×3], Dic5 [×2], Dic5 [×5], C20 [×6], D10 [×6], C2×C10, C2×C10 [×3], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C422C2, C422C2, C4⋊Q8, Dic10 [×4], C4×D5 [×3], D20, C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×3], C2×C20 [×6], C22×D5 [×2], C22×C10, C22.36C24, C4×Dic5 [×3], C10.D4 [×4], C4⋊Dic5 [×3], D10⋊C4 [×8], C23.D5, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5 [×2], C2×D20, C22×Dic5, C2×C5⋊D4 [×2], C4×Dic10, C4.D20, Dic5.14D4, Dic54D4, D10⋊D4, Dic5.5D4 [×2], C22.D20, C20⋊Q8, C4⋊C47D5, D10.13D4, D10⋊Q8, D102Q8, C4⋊C4⋊D5, C5×C422C2, C42.164D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.36C24, C23×D5, D5×C4○D4, D48D10, D4.10D10, C42.164D10

Smallest permutation representation of C42.164D10
On 160 points
Generators in S160
```(1 140 109 32)(2 131 110 23)(3 122 111 34)(4 133 112 25)(5 124 113 36)(6 135 114 27)(7 126 115 38)(8 137 116 29)(9 128 117 40)(10 139 118 31)(11 130 119 22)(12 121 120 33)(13 132 101 24)(14 123 102 35)(15 134 103 26)(16 125 104 37)(17 136 105 28)(18 127 106 39)(19 138 107 30)(20 129 108 21)(41 94 74 142)(42 85 75 153)(43 96 76 144)(44 87 77 155)(45 98 78 146)(46 89 79 157)(47 100 80 148)(48 91 61 159)(49 82 62 150)(50 93 63 141)(51 84 64 152)(52 95 65 143)(53 86 66 154)(54 97 67 145)(55 88 68 156)(56 99 69 147)(57 90 70 158)(58 81 71 149)(59 92 72 160)(60 83 73 151)
(1 50 11 60)(2 74 12 64)(3 52 13 42)(4 76 14 66)(5 54 15 44)(6 78 16 68)(7 56 17 46)(8 80 18 70)(9 58 19 48)(10 62 20 72)(21 92 31 82)(22 151 32 141)(23 94 33 84)(24 153 34 143)(25 96 35 86)(26 155 36 145)(27 98 37 88)(28 157 38 147)(29 100 39 90)(30 159 40 149)(41 120 51 110)(43 102 53 112)(45 104 55 114)(47 106 57 116)(49 108 59 118)(61 117 71 107)(63 119 73 109)(65 101 75 111)(67 103 77 113)(69 105 79 115)(81 138 91 128)(83 140 93 130)(85 122 95 132)(87 124 97 134)(89 126 99 136)(121 152 131 142)(123 154 133 144)(125 156 135 146)(127 158 137 148)(129 160 139 150)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 140 31 130)(22 129 32 139)(23 138 33 128)(24 127 34 137)(25 136 35 126)(26 125 36 135)(27 134 37 124)(28 123 38 133)(29 132 39 122)(30 121 40 131)(41 48 51 58)(42 57 52 47)(43 46 53 56)(44 55 54 45)(49 60 59 50)(61 64 71 74)(62 73 72 63)(65 80 75 70)(66 69 76 79)(67 78 77 68)(81 142 91 152)(82 151 92 141)(83 160 93 150)(84 149 94 159)(85 158 95 148)(86 147 96 157)(87 156 97 146)(88 145 98 155)(89 154 99 144)(90 143 100 153)(101 106 111 116)(102 115 112 105)(103 104 113 114)(107 120 117 110)(108 109 118 119)```

`G:=sub<Sym(160)| (1,140,109,32)(2,131,110,23)(3,122,111,34)(4,133,112,25)(5,124,113,36)(6,135,114,27)(7,126,115,38)(8,137,116,29)(9,128,117,40)(10,139,118,31)(11,130,119,22)(12,121,120,33)(13,132,101,24)(14,123,102,35)(15,134,103,26)(16,125,104,37)(17,136,105,28)(18,127,106,39)(19,138,107,30)(20,129,108,21)(41,94,74,142)(42,85,75,153)(43,96,76,144)(44,87,77,155)(45,98,78,146)(46,89,79,157)(47,100,80,148)(48,91,61,159)(49,82,62,150)(50,93,63,141)(51,84,64,152)(52,95,65,143)(53,86,66,154)(54,97,67,145)(55,88,68,156)(56,99,69,147)(57,90,70,158)(58,81,71,149)(59,92,72,160)(60,83,73,151), (1,50,11,60)(2,74,12,64)(3,52,13,42)(4,76,14,66)(5,54,15,44)(6,78,16,68)(7,56,17,46)(8,80,18,70)(9,58,19,48)(10,62,20,72)(21,92,31,82)(22,151,32,141)(23,94,33,84)(24,153,34,143)(25,96,35,86)(26,155,36,145)(27,98,37,88)(28,157,38,147)(29,100,39,90)(30,159,40,149)(41,120,51,110)(43,102,53,112)(45,104,55,114)(47,106,57,116)(49,108,59,118)(61,117,71,107)(63,119,73,109)(65,101,75,111)(67,103,77,113)(69,105,79,115)(81,138,91,128)(83,140,93,130)(85,122,95,132)(87,124,97,134)(89,126,99,136)(121,152,131,142)(123,154,133,144)(125,156,135,146)(127,158,137,148)(129,160,139,150), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,140,31,130)(22,129,32,139)(23,138,33,128)(24,127,34,137)(25,136,35,126)(26,125,36,135)(27,134,37,124)(28,123,38,133)(29,132,39,122)(30,121,40,131)(41,48,51,58)(42,57,52,47)(43,46,53,56)(44,55,54,45)(49,60,59,50)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68)(81,142,91,152)(82,151,92,141)(83,160,93,150)(84,149,94,159)(85,158,95,148)(86,147,96,157)(87,156,97,146)(88,145,98,155)(89,154,99,144)(90,143,100,153)(101,106,111,116)(102,115,112,105)(103,104,113,114)(107,120,117,110)(108,109,118,119)>;`

`G:=Group( (1,140,109,32)(2,131,110,23)(3,122,111,34)(4,133,112,25)(5,124,113,36)(6,135,114,27)(7,126,115,38)(8,137,116,29)(9,128,117,40)(10,139,118,31)(11,130,119,22)(12,121,120,33)(13,132,101,24)(14,123,102,35)(15,134,103,26)(16,125,104,37)(17,136,105,28)(18,127,106,39)(19,138,107,30)(20,129,108,21)(41,94,74,142)(42,85,75,153)(43,96,76,144)(44,87,77,155)(45,98,78,146)(46,89,79,157)(47,100,80,148)(48,91,61,159)(49,82,62,150)(50,93,63,141)(51,84,64,152)(52,95,65,143)(53,86,66,154)(54,97,67,145)(55,88,68,156)(56,99,69,147)(57,90,70,158)(58,81,71,149)(59,92,72,160)(60,83,73,151), (1,50,11,60)(2,74,12,64)(3,52,13,42)(4,76,14,66)(5,54,15,44)(6,78,16,68)(7,56,17,46)(8,80,18,70)(9,58,19,48)(10,62,20,72)(21,92,31,82)(22,151,32,141)(23,94,33,84)(24,153,34,143)(25,96,35,86)(26,155,36,145)(27,98,37,88)(28,157,38,147)(29,100,39,90)(30,159,40,149)(41,120,51,110)(43,102,53,112)(45,104,55,114)(47,106,57,116)(49,108,59,118)(61,117,71,107)(63,119,73,109)(65,101,75,111)(67,103,77,113)(69,105,79,115)(81,138,91,128)(83,140,93,130)(85,122,95,132)(87,124,97,134)(89,126,99,136)(121,152,131,142)(123,154,133,144)(125,156,135,146)(127,158,137,148)(129,160,139,150), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,140,31,130)(22,129,32,139)(23,138,33,128)(24,127,34,137)(25,136,35,126)(26,125,36,135)(27,134,37,124)(28,123,38,133)(29,132,39,122)(30,121,40,131)(41,48,51,58)(42,57,52,47)(43,46,53,56)(44,55,54,45)(49,60,59,50)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68)(81,142,91,152)(82,151,92,141)(83,160,93,150)(84,149,94,159)(85,158,95,148)(86,147,96,157)(87,156,97,146)(88,145,98,155)(89,154,99,144)(90,143,100,153)(101,106,111,116)(102,115,112,105)(103,104,113,114)(107,120,117,110)(108,109,118,119) );`

`G=PermutationGroup([(1,140,109,32),(2,131,110,23),(3,122,111,34),(4,133,112,25),(5,124,113,36),(6,135,114,27),(7,126,115,38),(8,137,116,29),(9,128,117,40),(10,139,118,31),(11,130,119,22),(12,121,120,33),(13,132,101,24),(14,123,102,35),(15,134,103,26),(16,125,104,37),(17,136,105,28),(18,127,106,39),(19,138,107,30),(20,129,108,21),(41,94,74,142),(42,85,75,153),(43,96,76,144),(44,87,77,155),(45,98,78,146),(46,89,79,157),(47,100,80,148),(48,91,61,159),(49,82,62,150),(50,93,63,141),(51,84,64,152),(52,95,65,143),(53,86,66,154),(54,97,67,145),(55,88,68,156),(56,99,69,147),(57,90,70,158),(58,81,71,149),(59,92,72,160),(60,83,73,151)], [(1,50,11,60),(2,74,12,64),(3,52,13,42),(4,76,14,66),(5,54,15,44),(6,78,16,68),(7,56,17,46),(8,80,18,70),(9,58,19,48),(10,62,20,72),(21,92,31,82),(22,151,32,141),(23,94,33,84),(24,153,34,143),(25,96,35,86),(26,155,36,145),(27,98,37,88),(28,157,38,147),(29,100,39,90),(30,159,40,149),(41,120,51,110),(43,102,53,112),(45,104,55,114),(47,106,57,116),(49,108,59,118),(61,117,71,107),(63,119,73,109),(65,101,75,111),(67,103,77,113),(69,105,79,115),(81,138,91,128),(83,140,93,130),(85,122,95,132),(87,124,97,134),(89,126,99,136),(121,152,131,142),(123,154,133,144),(125,156,135,146),(127,158,137,148),(129,160,139,150)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,140,31,130),(22,129,32,139),(23,138,33,128),(24,127,34,137),(25,136,35,126),(26,125,36,135),(27,134,37,124),(28,123,38,133),(29,132,39,122),(30,121,40,131),(41,48,51,58),(42,57,52,47),(43,46,53,56),(44,55,54,45),(49,60,59,50),(61,64,71,74),(62,73,72,63),(65,80,75,70),(66,69,76,79),(67,78,77,68),(81,142,91,152),(82,151,92,141),(83,160,93,150),(84,149,94,159),(85,158,95,148),(86,147,96,157),(87,156,97,146),(88,145,98,155),(89,154,99,144),(90,143,100,153),(101,106,111,116),(102,115,112,105),(103,104,113,114),(107,120,117,110),(108,109,118,119)])`

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L 4M 4N 4O 5A 5B 10A ··· 10F 10G 10H 20A ··· 20L 20M ··· 20R order 1 2 2 2 2 2 2 4 4 4 ··· 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 20 20 2 2 4 ··· 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 8 8 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2+ 1+4 2- 1+4 D5×C4○D4 D4⋊8D10 D4.10D10 kernel C42.164D10 C4×Dic10 C4.D20 Dic5.14D4 Dic5⋊4D4 D10⋊D4 Dic5.5D4 C22.D20 C20⋊Q8 C4⋊C4⋊7D5 D10.13D4 D10⋊Q8 D10⋊2Q8 C4⋊C4⋊D5 C5×C42⋊2C2 C42⋊2C2 Dic5 C42 C22⋊C4 C4⋊C4 C10 C10 C2 C2 C2 # reps 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 4 2 6 6 1 1 4 4 4

Matrix representation of C42.164D10 in GL6(𝔽41)

 32 39 0 0 0 0 40 9 0 0 0 0 0 0 2 13 0 0 0 0 28 39 0 0 0 0 0 0 2 13 0 0 0 0 28 39
,
 40 18 0 0 0 0 9 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 9 0 0 0 0 0 1 32 0 0 0 0 0 0 32 32 39 39 0 0 9 19 2 27 0 0 2 2 9 9 0 0 39 14 32 22
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 32 32 39 39 0 0 19 9 27 2 0 0 2 2 9 9 0 0 14 39 22 32

`G:=sub<GL(6,GF(41))| [32,40,0,0,0,0,39,9,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,9,0,0,0,0,18,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[9,1,0,0,0,0,0,32,0,0,0,0,0,0,32,9,2,39,0,0,32,19,2,14,0,0,39,2,9,32,0,0,39,27,9,22],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,19,2,14,0,0,32,9,2,39,0,0,39,27,9,22,0,0,39,2,9,32] >;`

C42.164D10 in GAP, Magma, Sage, TeX

`C_4^2._{164}D_{10}`
`% in TeX`

`G:=Group("C4^2.164D10");`
`// GroupNames label`

`G:=SmallGroup(320,1382);`
`// by ID`

`G=gap.SmallGroup(320,1382);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,268,675,570,80,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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