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## G = C42.95D10order 320 = 26·5

### 95th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.95D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.95D10
 Lower central C5 — C2×C10 — C42.95D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.95D10
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c9 >

Subgroups: 902 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×13], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×10], C23, C23 [×3], D5 [×4], C10 [×3], C10, C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×5], C2×D4 [×6], Dic5 [×5], C20 [×2], C20 [×5], D10 [×2], D10 [×8], C2×C10, C2×C10 [×3], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C422C2 [×2], C4×D5 [×6], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5, C22×D5 [×2], C22×C10, C22.47C24, C4×Dic5, C10.D4, C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×6], C23.D5, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20, C2×D20 [×2], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, C4×D20 [×2], C422D5 [×2], D10.12D4 [×2], D10⋊D4 [×2], C4.Dic10, D5×C4⋊C4, D208C4, C4⋊D20, C4×C5⋊D4, C207D4, C5×C42⋊C2, C42.95D10
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], C22.47C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D48D10, C42.95D10

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

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20AB order 1 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 4 10 10 20 20 2 ··· 2 4 4 10 10 20 20 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⋊8D10 kernel C42.95D10 C4×D20 C42⋊2D5 D10.12D4 D10⋊D4 C4.Dic10 D5×C4⋊C4 D20⋊8C4 C4⋊D20 C4×C5⋊D4 C20⋊7D4 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 C42.95D10 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 32 0 0 0 20 9
,
 24 1 0 0 40 17 0 0 0 0 9 0 0 0 0 9
,
 34 34 0 0 7 1 0 0 0 0 40 36 0 0 25 1
,
 14 27 0 0 11 27 0 0 0 0 40 36 0 0 25 1
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,32,20,0,0,0,9],[24,40,0,0,1,17,0,0,0,0,9,0,0,0,0,9],[34,7,0,0,34,1,0,0,0,0,40,25,0,0,36,1],[14,11,0,0,27,27,0,0,0,0,40,25,0,0,36,1] >;`

C42.95D10 in GAP, Magma, Sage, TeX

`C_4^2._{95}D_{10}`
`% in TeX`

`G:=Group("C4^2.95D10");`
`// GroupNames label`

`G:=SmallGroup(320,1202);`
`// by ID`

`G=gap.SmallGroup(320,1202);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,136,12550]);`
`// 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^-1,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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