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## G = C42.166D10order 320 = 26·5

### 166th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.166D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — D4×Dic5 — C42.166D10
 Lower central C5 — C2×C10 — C42.166D10
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for C42.166D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 726 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×10], Q8 [×4], C23 [×4], C10, C10 [×2], C10 [×4], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], Dic5 [×8], C20 [×4], C20, C2×C10, C2×C10 [×12], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C41D4, Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C5×D4 [×10], C22×C10 [×4], C22.53C24, C4×Dic5 [×4], C10.D4 [×4], C4⋊Dic5 [×2], C23.D5 [×12], C4×C20, C2×Dic10 [×2], C22×Dic5 [×4], D4×C10 [×6], C4×Dic10 [×2], D4×Dic5 [×4], C23.18D10 [×4], C20.17D4 [×4], C5×C41D4, C42.166D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.53C24, D42D5 [×4], C23×D5, C2×D42D5 [×2], D46D10, C42.166D10

Smallest permutation representation of C42.166D10
On 160 points
Generators in S160
```(1 54 59 6)(2 7 60 55)(3 56 51 8)(4 9 52 57)(5 58 53 10)(11 83 23 75)(12 76 24 84)(13 85 25 77)(14 78 26 86)(15 87 27 79)(16 80 28 88)(17 89 29 71)(18 72 30 90)(19 81 21 73)(20 74 22 82)(31 36 64 69)(32 70 65 37)(33 38 66 61)(34 62 67 39)(35 40 68 63)(41 122 158 105)(42 106 159 123)(43 124 160 107)(44 108 151 125)(45 126 152 109)(46 110 153 127)(47 128 154 101)(48 102 155 129)(49 130 156 103)(50 104 157 121)(91 138 133 96)(92 97 134 139)(93 140 135 98)(94 99 136 131)(95 132 137 100)(111 148 143 116)(112 117 144 149)(113 150 145 118)(114 119 146 141)(115 142 147 120)
(1 73 63 26)(2 27 64 74)(3 75 65 28)(4 29 66 76)(5 77 67 30)(6 21 68 78)(7 79 69 22)(8 23 70 80)(9 71 61 24)(10 25 62 72)(11 37 88 56)(12 57 89 38)(13 39 90 58)(14 59 81 40)(15 31 82 60)(16 51 83 32)(17 33 84 52)(18 53 85 34)(19 35 86 54)(20 55 87 36)(41 148 127 100)(42 91 128 149)(43 150 129 92)(44 93 130 141)(45 142 121 94)(46 95 122 143)(47 144 123 96)(48 97 124 145)(49 146 125 98)(50 99 126 147)(101 117 159 133)(102 134 160 118)(103 119 151 135)(104 136 152 120)(105 111 153 137)(106 138 154 112)(107 113 155 139)(108 140 156 114)(109 115 157 131)(110 132 158 116)
(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)
(1 146 63 98)(2 145 64 97)(3 144 65 96)(4 143 66 95)(5 142 67 94)(6 141 68 93)(7 150 69 92)(8 149 70 91)(9 148 61 100)(10 147 62 99)(11 42 88 128)(12 41 89 127)(13 50 90 126)(14 49 81 125)(15 48 82 124)(16 47 83 123)(17 46 84 122)(18 45 85 121)(19 44 86 130)(20 43 87 129)(21 151 78 103)(22 160 79 102)(23 159 80 101)(24 158 71 110)(25 157 72 109)(26 156 73 108)(27 155 74 107)(28 154 75 106)(29 153 76 105)(30 152 77 104)(31 139 60 113)(32 138 51 112)(33 137 52 111)(34 136 53 120)(35 135 54 119)(36 134 55 118)(37 133 56 117)(38 132 57 116)(39 131 58 115)(40 140 59 114)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F ··· 4M 4N 4O 4P 4Q 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20L order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 4 4 4 4 2 2 2 2 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 8 ··· 8 4 ··· 4

53 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 2+ 1+4 D4⋊2D5 D4⋊6D10 kernel C42.166D10 C4×Dic10 D4×Dic5 C23.18D10 C20.17D4 C5×C4⋊1D4 C4⋊1D4 C20 C42 C2×D4 C10 C4 C2 # reps 1 2 4 4 4 1 2 8 2 12 1 8 4

Matrix representation of C42.166D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 35 0 0 0 0 6 35 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 2 25 0 0 0 0 13 39 0 0 0 0 0 0 0 9 0 0 0 0 9 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,2,13,0,0,0,0,25,39,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;`

C42.166D10 in GAP, Magma, Sage, TeX

`C_4^2._{166}D_{10}`
`% in TeX`

`G:=Group("C4^2.166D10");`
`// GroupNames label`

`G:=SmallGroup(320,1385);`
`// by ID`

`G=gap.SmallGroup(320,1385);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,1571,570,297,136,12550]);`
`// Polycyclic`

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

׿
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