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## G = C42.133D10order 320 = 26·5

### 133rd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.133D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C42⋊D5 — C42.133D10
 Lower central C5 — C2×C10 — C42.133D10
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.133D10
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c9 >

Subgroups: 814 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×4], Q8 [×4], C23 [×3], D5 [×3], C10 [×3], C42, C42 [×2], C42, C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2×Q8 [×2], Dic5 [×5], C20 [×2], C20 [×6], D10 [×9], C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C422C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22.36C24, C4×Dic5, C10.D4 [×2], C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×2], D10⋊C4 [×10], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q8×C10, C42⋊D5, C4×D20, C4.D20 [×2], C422D5 [×2], C20⋊Q8, D10.13D4 [×2], C4⋊D20, D10⋊Q8 [×2], D103Q8, C20.23D4, Q8×C20, C42.133D10
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, C4○D20 [×2], C23×D5, C2×C4○D20, Q8.10D10, D48D10, C42.133D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4F 4G 4H 4I 4J 4K ··· 4O 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 20 20 20 2 ··· 2 4 4 4 4 20 ··· 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 C4○D20 2+ 1+4 2- 1+4 Q8.10D10 D4⋊8D10 kernel C42.133D10 C42⋊D5 C4×D20 C4.D20 C42⋊2D5 C20⋊Q8 D10.13D4 C4⋊D20 D10⋊Q8 D10⋊3Q8 C20.23D4 Q8×C20 C4×Q8 C20 C42 C4⋊C4 C2×Q8 C4 C10 C10 C2 C2 # reps 1 1 1 2 2 1 2 1 2 1 1 1 2 4 6 6 2 16 1 1 4 4

Matrix representation of C42.133D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 30 14 31 14 0 0 10 3 10 4 0 0 33 15 11 27 0 0 40 7 31 38
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 11 32 0 0 0 0 9 30 0 0 0 0 28 0 39 32 0 0 28 13 37 2
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 19 9 37 13 0 0 4 4 0 9 0 0 2 15 5 32 0 0 40 17 20 13
,
 9 0 0 0 0 0 32 32 0 0 0 0 0 0 35 4 5 36 0 0 3 17 30 6 0 0 31 23 20 32 0 0 24 40 38 10

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,10,33,40,0,0,14,3,15,7,0,0,31,10,11,31,0,0,14,4,27,38],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,11,9,28,28,0,0,32,30,0,13,0,0,0,0,39,37,0,0,0,0,32,2],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,19,4,2,40,0,0,9,4,15,17,0,0,37,0,5,20,0,0,13,9,32,13],[9,32,0,0,0,0,0,32,0,0,0,0,0,0,35,3,31,24,0,0,4,17,23,40,0,0,5,30,20,38,0,0,36,6,32,10] >;`

C42.133D10 in GAP, Magma, Sage, TeX

`C_4^2._{133}D_{10}`
`% in TeX`

`G:=Group("C4^2.133D10");`
`// GroupNames label`

`G:=SmallGroup(320,1254);`
`// by ID`

`G=gap.SmallGroup(320,1254);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,136,12550]);`
`// Polycyclic`

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

׿
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