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G = C42.188D10order 320 = 26·5

8th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.188D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×C4○D20 — C42.188D10
 Lower central C5 — C10 — C42.188D10
 Upper central C1 — C2×C4 — C42⋊C2

Generators and relations for C42.188D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 926 in 310 conjugacy classes, 155 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×14], C22, C22 [×2], C22 [×10], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×26], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C42 [×2], C42 [×8], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×8], Dic5 [×2], C20 [×4], C20 [×4], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42 [×3], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], C4×D5 [×8], D20 [×4], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×C10, C4×C4○D4, C4×Dic5 [×2], C4×Dic5 [×6], C10.D4 [×4], D10⋊C4 [×4], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×6], C2×D20, C4○D20 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20, D5×C42 [×2], C42⋊D5 [×2], Dic54D4 [×4], Dic53Q8 [×2], D208C4 [×2], C2×C4×Dic5, C5×C42⋊C2, C2×C4○D20, C42.188D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C4×C4○D4, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D5×C4○D4 [×2], C42.188D10

Smallest permutation representation of C42.188D10
On 160 points
Generators in S160
```(1 143 63 98)(2 144 64 99)(3 145 65 100)(4 146 66 91)(5 147 67 92)(6 148 68 93)(7 149 69 94)(8 150 70 95)(9 141 61 96)(10 142 62 97)(11 43 87 123)(12 44 88 124)(13 45 89 125)(14 46 90 126)(15 47 81 127)(16 48 82 128)(17 49 83 129)(18 50 84 130)(19 41 85 121)(20 42 86 122)(21 156 76 101)(22 157 77 102)(23 158 78 103)(24 159 79 104)(25 160 80 105)(26 151 71 106)(27 152 72 107)(28 153 73 108)(29 154 74 109)(30 155 75 110)(31 132 52 111)(32 133 53 112)(33 134 54 113)(34 135 55 114)(35 136 56 115)(36 137 57 116)(37 138 58 117)(38 139 59 118)(39 140 60 119)(40 131 51 120)
(1 14 51 28)(2 81 52 74)(3 16 53 30)(4 83 54 76)(5 18 55 22)(6 85 56 78)(7 20 57 24)(8 87 58 80)(9 12 59 26)(10 89 60 72)(11 37 25 70)(13 39 27 62)(15 31 29 64)(17 33 21 66)(19 35 23 68)(32 75 65 82)(34 77 67 84)(36 79 69 86)(38 71 61 88)(40 73 63 90)(41 136 158 93)(42 116 159 149)(43 138 160 95)(44 118 151 141)(45 140 152 97)(46 120 153 143)(47 132 154 99)(48 112 155 145)(49 134 156 91)(50 114 157 147)(92 130 135 102)(94 122 137 104)(96 124 139 106)(98 126 131 108)(100 128 133 110)(101 146 129 113)(103 148 121 115)(105 150 123 117)(107 142 125 119)(109 144 127 111)
(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)
(1 135 63 114)(2 113 64 134)(3 133 65 112)(4 111 66 132)(5 131 67 120)(6 119 68 140)(7 139 69 118)(8 117 70 138)(9 137 61 116)(10 115 62 136)(11 105 87 160)(12 159 88 104)(13 103 89 158)(14 157 90 102)(15 101 81 156)(16 155 82 110)(17 109 83 154)(18 153 84 108)(19 107 85 152)(20 151 86 106)(21 127 76 47)(22 46 77 126)(23 125 78 45)(24 44 79 124)(25 123 80 43)(26 42 71 122)(27 121 72 41)(28 50 73 130)(29 129 74 49)(30 48 75 128)(31 91 52 146)(32 145 53 100)(33 99 54 144)(34 143 55 98)(35 97 56 142)(36 141 57 96)(37 95 58 150)(38 149 59 94)(39 93 60 148)(40 147 51 92)```

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4N 4O ··· 4V 4W ··· 4AD 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20AB order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 1 1 1 1 2 ··· 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D5 C4○D4 D10 D10 D10 D10 C4×D5 D5×C4○D4 kernel C42.188D10 D5×C42 C42⋊D5 Dic5⋊4D4 Dic5⋊3Q8 D20⋊8C4 C2×C4×Dic5 C5×C42⋊C2 C2×C4○D20 C4○D20 C42⋊C2 Dic5 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C2 # reps 1 2 2 4 2 2 1 1 1 16 2 8 4 4 4 2 16 8

Matrix representation of C42.188D10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 32 0 0 0 0 32
,
 9 0 0 0 0 9 0 0 0 0 1 21 0 0 0 40
,
 6 6 0 0 35 1 0 0 0 0 9 25 0 0 5 32
,
 6 6 0 0 1 35 0 0 0 0 9 0 0 0 5 32
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[9,0,0,0,0,9,0,0,0,0,1,0,0,0,21,40],[6,35,0,0,6,1,0,0,0,0,9,5,0,0,25,32],[6,1,0,0,6,35,0,0,0,0,9,5,0,0,0,32] >;`

C42.188D10 in GAP, Magma, Sage, TeX

`C_4^2._{188}D_{10}`
`% in TeX`

`G:=Group("C4^2.188D10");`
`// GroupNames label`

`G:=SmallGroup(320,1194);`
`// by ID`

`G=gap.SmallGroup(320,1194);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,297,80,12550]);`
`// Polycyclic`

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

׿
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