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## G = C42.132D10order 320 = 26·5

### 132nd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.132D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.132D10
 Lower central C5 — C2×C10 — C42.132D10
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.132D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 694 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×7], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×4], C2×D4, C2×Q8, Dic5 [×6], C20 [×2], C20 [×6], D10 [×2], D10 [×5], C2×C10, C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C422C2 [×2], C4×D5 [×8], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], C22.46C24, C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5 [×3], D10⋊C4 [×2], D10⋊C4 [×6], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, Q8×C10, C20.6Q8, C42⋊D5 [×2], C4×D20, C422D5 [×2], Dic5.Q8 [×2], D5×C4⋊C4, C4⋊C47D5, D10.13D4 [×2], D103Q8 [×2], Q8×C20, C42.132D10
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], C22.46C24, C4○D20 [×2], C23×D5, C2×C4○D20, Q8.10D10, D5×C4○D4, C42.132D10

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 C4○D20 2- 1+4 Q8.10D10 D5×C4○D4 kernel C42.132D10 C20.6Q8 C42⋊D5 C4×D20 C42⋊2D5 Dic5.Q8 D5×C4⋊C4 C4⋊C4⋊7D5 D10.13D4 D10⋊3Q8 Q8×C20 C4×Q8 C20 D10 C42 C4⋊C4 C2×Q8 C4 C10 C2 C2 # reps 1 1 2 1 2 2 1 1 2 2 1 2 4 4 6 6 2 16 1 4 4

Matrix representation of C42.132D10 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 32
,
 39 28 0 0 13 2 0 0 0 0 9 0 0 0 0 9
,
 28 28 0 0 13 32 0 0 0 0 0 1 0 0 1 0
,
 20 21 0 0 18 21 0 0 0 0 0 40 0 0 40 0
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[39,13,0,0,28,2,0,0,0,0,9,0,0,0,0,9],[28,13,0,0,28,32,0,0,0,0,0,1,0,0,1,0],[20,18,0,0,21,21,0,0,0,0,0,40,0,0,40,0] >;`

C42.132D10 in GAP, Magma, Sage, TeX

`C_4^2._{132}D_{10}`
`% in TeX`

`G:=Group("C4^2.132D10");`
`// GroupNames label`

`G:=SmallGroup(320,1253);`
`// by ID`

`G=gap.SmallGroup(320,1253);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,1571,136,12550]);`
`// Polycyclic`

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

׿
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