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## G = C42.91D10order 320 = 26·5

### 91st non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.91D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×C4○D20 — C42.91D10
 Lower central C5 — C10 — C42.91D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.91D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=c9 >

Subgroups: 926 in 294 conjugacy classes, 151 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C23.33C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, C4×Dic10, C4×D20, Dic54D4, D5×C4⋊C4, C4⋊C47D5, C2×C4⋊Dic5, C5×C42⋊C2, C2×C4○D20, C42.91D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2+ 1+4, 2- 1+4, C4×D5, C22×D5, C23.33C23, C2×C4×D5, C23×D5, D5×C22×C4, D48D10, D4.10D10, C42.91D10

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4L 4M ··· 4X 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 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 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 C4 D5 D10 D10 D10 D10 C4×D5 2+ 1+4 2- 1+4 D4⋊8D10 D4.10D10 kernel C42.91D10 C4×Dic10 C4×D20 Dic5⋊4D4 D5×C4⋊C4 C4⋊C4⋊7D5 C2×C4⋊Dic5 C5×C42⋊C2 C2×C4○D20 C4○D20 C42⋊C2 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C10 C10 C2 C2 # reps 1 2 2 4 2 2 1 1 1 16 2 4 4 4 2 16 1 1 4 4

Matrix representation of C42.91D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 4 0 34 34 0 0 0 4 14 1 0 0 30 5 37 0 0 0 31 36 0 37
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 38 30 18 23 0 0 6 3 0 21 0 0 26 34 33 11 0 0 0 34 30 8
,
 7 6 0 0 0 0 34 0 0 0 0 0 0 0 6 11 3 1 0 0 35 0 37 37 0 0 17 21 5 30 0 0 13 34 11 30
,
 34 40 0 0 0 0 7 7 0 0 0 0 0 0 12 10 22 9 0 0 25 29 19 10 0 0 29 19 10 17 0 0 31 31 39 31

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,4,0,30,31,0,0,0,4,5,36,0,0,34,14,37,0,0,0,34,1,0,37],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,38,6,26,0,0,0,30,3,34,34,0,0,18,0,33,30,0,0,23,21,11,8],[7,34,0,0,0,0,6,0,0,0,0,0,0,0,6,35,17,13,0,0,11,0,21,34,0,0,3,37,5,11,0,0,1,37,30,30],[34,7,0,0,0,0,40,7,0,0,0,0,0,0,12,25,29,31,0,0,10,29,19,31,0,0,22,19,10,39,0,0,9,10,17,31] >;`

C42.91D10 in GAP, Magma, Sage, TeX

`C_4^2._{91}D_{10}`
`% in TeX`

`G:=Group("C4^2.91D10");`
`// GroupNames label`

`G:=SmallGroup(320,1195);`
`// by ID`

`G=gap.SmallGroup(320,1195);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,570,297,80,12550]);`
`// 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^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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