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## G = C42.131D10order 320 = 26·5

### 131st non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.131D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.131D10
 Lower central C5 — C2×C10 — C42.131D10
 Upper central C1 — C2×C4 — C4×Q8

Generators and relations for C42.131D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 814 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×3], D5 [×4], C10 [×3], C42, C42 [×2], C42 [×3], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×5], C20 [×4], C20 [×5], D10 [×2], D10 [×8], C2×C10, C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], C4×D5 [×10], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C23.36C23, C4×Dic5 [×3], C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C4×D5 [×3], C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q8×C10, D5×C42, C42⋊D5 [×2], C4×D20, C4×D20 [×2], C4.Dic10, D10.13D4 [×2], C4⋊D20, C4⋊C4⋊D5 [×2], D103Q8, C20.23D4, Q8×C20, C42.131D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, C4○D20 [×2], Q82D5 [×2], C23×D5, C2×C4○D20, C2×Q82D5, D5×C4○D4, C42.131D10

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

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

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 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 Q8⋊2D5 D5×C4○D4 kernel C42.131D10 D5×C42 C42⋊D5 C4×D20 C4.Dic10 D10.13D4 C4⋊D20 C4⋊C4⋊D5 D10⋊3Q8 C20.23D4 Q8×C20 C4×Q8 C20 D10 C42 C4⋊C4 C2×Q8 C4 C4 C2 # reps 1 1 2 3 1 2 1 2 1 1 1 2 8 4 6 6 2 16 4 4

Matrix representation of C42.131D10 in GL4(𝔽41) generated by

 25 15 0 0 2 16 0 0 0 0 18 6 0 0 35 23
,
 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 25 16 0 0 2 16 0 0 0 0 9 13 0 0 28 13
,
 16 25 0 0 39 25 0 0 0 0 23 35 0 0 20 18
`G:=sub<GL(4,GF(41))| [25,2,0,0,15,16,0,0,0,0,18,35,0,0,6,23],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[25,2,0,0,16,16,0,0,0,0,9,28,0,0,13,13],[16,39,0,0,25,25,0,0,0,0,23,20,0,0,35,18] >;`

C42.131D10 in GAP, Magma, Sage, TeX

`C_4^2._{131}D_{10}`
`% in TeX`

`G:=Group("C4^2.131D10");`
`// GroupNames label`

`G:=SmallGroup(320,1252);`
`// by ID`

`G=gap.SmallGroup(320,1252);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,794,136,12550]);`
`// 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=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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