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## G = C42.108D10order 320 = 26·5

### 108th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.108D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×D4⋊2D5 — C42.108D10
 Lower central C5 — C10 — C42.108D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.108D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 862 in 294 conjugacy classes, 151 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×14], C22, C22 [×4], C22 [×8], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×25], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×4], C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×6], Dic5 [×4], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4, C4×D4 [×5], C4×Q8 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×12], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×C10 [×2], C23.33C23, C4×Dic5, C4×Dic5 [×4], C10.D4 [×2], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], D42D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C42⋊D5, C23.11D10 [×2], Dic54D4 [×2], Dic53Q8, D5×C4⋊C4, C2×C10.D4 [×2], C4×C5⋊D4 [×2], D4×Dic5, D4×C20, C2×D42D5, C42.108D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ 1+4, 2- 1+4, C4×D5 [×4], C22×D5 [×7], C23.33C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D46D10, D4.10D10, C42.108D10

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4J 4K ··· 4X 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 10 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 D5 D10 D10 D10 D10 D10 C4×D5 2+ 1+4 2- 1+4 D4⋊6D10 D4.10D10 kernel C42.108D10 C4×Dic10 C42⋊D5 C23.11D10 Dic5⋊4D4 Dic5⋊3Q8 D5×C4⋊C4 C2×C10.D4 C4×C5⋊D4 D4×Dic5 D4×C20 C2×D4⋊2D5 D4⋊2D5 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C10 C10 C2 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 16 2 2 4 2 4 2 16 1 1 4 4

Matrix representation of C42.108D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 22 0 0 0 0 40 0 22 0 0 26 0 1 0 0 0 0 26 0 1
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 17 1 0 0 0 0 40 24 0 0 0 0 0 0 17 1 0 0 0 0 40 24
,
 0 34 0 0 0 0 6 35 0 0 0 0 0 0 5 25 26 7 0 0 16 16 34 34 0 0 7 35 36 16 0 0 6 6 25 25
,
 6 34 0 0 0 0 5 35 0 0 0 0 0 0 25 5 7 26 0 0 16 16 34 34 0 0 35 7 16 36 0 0 6 6 25 25

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,26,0,0,0,0,40,0,26,0,0,22,0,1,0,0,0,0,22,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,17,40,0,0,0,0,1,24,0,0,0,0,0,0,17,40,0,0,0,0,1,24],[0,6,0,0,0,0,34,35,0,0,0,0,0,0,5,16,7,6,0,0,25,16,35,6,0,0,26,34,36,25,0,0,7,34,16,25],[6,5,0,0,0,0,34,35,0,0,0,0,0,0,25,16,35,6,0,0,5,16,7,6,0,0,7,34,16,25,0,0,26,34,36,25] >;`

C42.108D10 in GAP, Magma, Sage, TeX

`C_4^2._{108}D_{10}`
`% in TeX`

`G:=Group("C4^2.108D10");`
`// GroupNames label`

`G:=SmallGroup(320,1218);`
`// by ID`

`G=gap.SmallGroup(320,1218);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,570,80,12550]);`
`// Polycyclic`

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

׿
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