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## G = C42.94D10order 320 = 26·5

### 94th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.94D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.94D10
 Lower central C5 — C2×C10 — C42.94D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.94D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 662 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×7], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D5 [×2], C10 [×3], C10, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic5 [×7], C20 [×2], C20 [×5], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C2×C4⋊C4, C42⋊C2, C42⋊C2 [×2], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C422C2 [×2], Dic10 [×2], C4×D5 [×6], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5, C22×C10, C22.46C24, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×8], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4, D10⋊C4 [×2], C23.D5, C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C2×C5⋊D4, C22×C20, C20.6Q8 [×2], C42⋊D5 [×2], C23.D10 [×2], D10.12D4 [×2], Dic53Q8, C4.Dic10, D5×C4⋊C4, D102Q8, C20.48D4, C4×C5⋊D4, C5×C42⋊C2, C42.94D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.46C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4.10D10, C42.94D10

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 C4○D20 2- 1+4 D5×C4○D4 D4.10D10 kernel C42.94D10 C20.6Q8 C42⋊D5 C23.D10 D10.12D4 Dic5⋊3Q8 C4.Dic10 D5×C4⋊C4 D10⋊2Q8 C20.48D4 C4×C5⋊D4 C5×C42⋊C2 C42⋊C2 C20 D10 C42 C22⋊C4 C4⋊C4 C22×C4 C4 C10 C2 C2 # reps 1 2 2 2 2 1 1 1 1 1 1 1 2 4 4 4 4 4 2 16 1 4 4

Matrix representation of C42.94D10 in GL4(𝔽41) generated by

 34 39 0 0 24 7 0 0 0 0 9 0 0 0 0 9
,
 32 0 0 0 0 32 0 0 0 0 23 35 0 0 6 18
,
 13 35 0 0 1 28 0 0 0 0 40 35 0 0 6 35
,
 28 6 0 0 40 13 0 0 0 0 40 0 0 0 6 1
`G:=sub<GL(4,GF(41))| [34,24,0,0,39,7,0,0,0,0,9,0,0,0,0,9],[32,0,0,0,0,32,0,0,0,0,23,6,0,0,35,18],[13,1,0,0,35,28,0,0,0,0,40,6,0,0,35,35],[28,40,0,0,6,13,0,0,0,0,40,6,0,0,0,1] >;`

C42.94D10 in GAP, Magma, Sage, TeX

`C_4^2._{94}D_{10}`
`% in TeX`

`G:=Group("C4^2.94D10");`
`// GroupNames label`

`G:=SmallGroup(320,1201);`
`// by ID`

`G=gap.SmallGroup(320,1201);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,675,136,12550]);`
`// Polycyclic`

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

׿
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