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G = C42.233D10order 320 = 26·5

53rd non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.233D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×C4×D5 — D5×C42 — C42.233D10
 Lower central C5 — C2×C10 — C42.233D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.233D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c-1 >

Subgroups: 1166 in 310 conjugacy classes, 107 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×12], C22, C22 [×16], C5, C2×C4, C2×C4 [×4], C2×C4 [×21], D4 [×20], Q8 [×4], C23 [×2], C23 [×3], D5 [×4], C10, C10 [×2], C10 [×2], C42, C42 [×3], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×7], C2×D4, C2×D4 [×9], C2×Q8, C2×Q8, C4○D4 [×8], Dic5 [×6], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×8], C2×C10, C2×C10 [×6], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4 [×2], Dic10 [×2], C4×D5 [×4], C4×D5 [×8], D20 [×6], C2×Dic5, C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], C22×C10 [×2], C22.26C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×4], D10⋊C4 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], D42D5 [×4], Q82D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×6], D4×C10, Q8×C10, D5×C42, C4.D20, Dic54D4 [×4], D10⋊D4 [×4], C20⋊D4, Dic5⋊Q8, C5×C4.4D4, C2×D42D5, C2×Q82D5, C42.233D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.26C24, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C4○D4 [×2], C42.233D10

Smallest permutation representation of C42.233D10
On 160 points
Generators in S160
```(1 135 33 106)(2 114 34 84)(3 137 35 108)(4 116 36 86)(5 139 37 110)(6 118 38 88)(7 131 39 102)(8 120 40 90)(9 133 31 104)(10 112 32 82)(11 119 41 89)(12 132 42 103)(13 111 43 81)(14 134 44 105)(15 113 45 83)(16 136 46 107)(17 115 47 85)(18 138 48 109)(19 117 49 87)(20 140 50 101)(21 95 143 55)(22 79 144 64)(23 97 145 57)(24 71 146 66)(25 99 147 59)(26 73 148 68)(27 91 149 51)(28 75 150 70)(29 93 141 53)(30 77 142 62)(52 155 92 122)(54 157 94 124)(56 159 96 126)(58 151 98 128)(60 153 100 130)(61 156 76 123)(63 158 78 125)(65 160 80 127)(67 152 72 129)(69 154 74 121)
(1 100 15 73)(2 51 16 69)(3 92 17 75)(4 53 18 61)(5 94 19 77)(6 55 20 63)(7 96 11 79)(8 57 12 65)(9 98 13 71)(10 59 14 67)(21 140 158 118)(22 102 159 89)(23 132 160 120)(24 104 151 81)(25 134 152 112)(26 106 153 83)(27 136 154 114)(28 108 155 85)(29 138 156 116)(30 110 157 87)(31 58 43 66)(32 99 44 72)(33 60 45 68)(34 91 46 74)(35 52 47 70)(36 93 48 76)(37 54 49 62)(38 95 50 78)(39 56 41 64)(40 97 42 80)(82 147 105 129)(84 149 107 121)(86 141 109 123)(88 143 101 125)(90 145 103 127)(111 146 133 128)(113 148 135 130)(115 150 137 122)(117 142 139 124)(119 144 131 126)
(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)
(1 124)(2 156)(3 122)(4 154)(5 130)(6 152)(7 128)(8 160)(9 126)(10 158)(11 146)(12 23)(13 144)(14 21)(15 142)(16 29)(17 150)(18 27)(19 148)(20 25)(22 43)(24 41)(26 49)(28 47)(30 45)(31 159)(32 125)(33 157)(34 123)(35 155)(36 121)(37 153)(38 129)(39 151)(40 127)(42 145)(44 143)(46 141)(48 149)(50 147)(51 86)(52 115)(53 84)(54 113)(55 82)(56 111)(57 90)(58 119)(59 88)(60 117)(61 107)(62 135)(63 105)(64 133)(65 103)(66 131)(67 101)(68 139)(69 109)(70 137)(71 102)(72 140)(73 110)(74 138)(75 108)(76 136)(77 106)(78 134)(79 104)(80 132)(81 96)(83 94)(85 92)(87 100)(89 98)(91 116)(93 114)(95 112)(97 120)(99 118)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 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 D4 D5 C4○D4 D10 D10 D10 D10 D4×D5 D5×C4○D4 kernel C42.233D10 D5×C42 C4.D20 Dic5⋊4D4 D10⋊D4 C20⋊D4 Dic5⋊Q8 C5×C4.4D4 C2×D4⋊2D5 C2×Q8⋊2D5 C4×D5 C4.4D4 Dic5 C42 C22⋊C4 C2×D4 C2×Q8 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 4 2 8 2 8 2 2 4 8

Matrix representation of C42.233D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 4 0 0 0 0 20 40
,
 9 18 0 0 0 0 32 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 37 0 0 0 0 21 1
,
 1 0 0 0 0 0 40 40 0 0 0 0 0 0 34 34 0 0 0 0 7 1 0 0 0 0 0 0 40 37 0 0 0 0 0 1
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 7 7 0 0 0 0 40 34 0 0 0 0 0 0 1 0 0 0 0 0 20 40

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,20,0,0,0,0,4,40],[9,32,0,0,0,0,18,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,21,0,0,0,0,37,1],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,0,0,0,0,0,37,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,1,20,0,0,0,0,0,40] >;`

C42.233D10 in GAP, Magma, Sage, TeX

`C_4^2._{233}D_{10}`
`% in TeX`

`G:=Group("C4^2.233D10");`
`// GroupNames label`

`G:=SmallGroup(320,1340);`
`// by ID`

`G=gap.SmallGroup(320,1340);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,232,100,1123,346,297,12550]);`
`// Polycyclic`

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

׿
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