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## G = C42.140D10order 320 = 26·5

### 140th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.140D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — Dic5.14D4 — C42.140D10
 Lower central C5 — C2×C10 — C42.140D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.140D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 614 in 196 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], C10, C10 [×2], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×8], C20 [×5], C2×C10, C2×C10 [×6], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C422C2 [×4], C4⋊Q8 [×2], Dic10 [×2], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20, C2×C20 [×4], C5×D4, C5×Q8, C22×C10 [×2], C22.57C24, C4×Dic5 [×2], C10.D4 [×12], C4⋊Dic5 [×4], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C22×Dic5 [×2], D4×C10, Q8×C10, C20.6Q8 [×2], Dic5.14D4 [×4], C23.D10 [×4], C23.18D10 [×2], Dic5⋊Q8 [×2], C5×C4.4D4, C42.140D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D5 [×7], C22.57C24, C23×D5, D46D10, D4.10D10 [×2], C42.140D10

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

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4E 4F ··· 4M 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 4 ··· 4 20 ··· 20 2 2 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

47 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 D10 2+ 1+4 2- 1+4 D4⋊6D10 D4.10D10 kernel C42.140D10 C20.6Q8 Dic5.14D4 C23.D10 C23.18D10 Dic5⋊Q8 C5×C4.4D4 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 C10 C10 C2 C2 # reps 1 2 4 4 2 2 1 2 2 8 2 2 1 2 4 8

Matrix representation of C42.140D10 in GL8(𝔽41)

 11 32 0 0 0 0 0 0 9 30 0 0 0 0 0 0 32 0 2 32 0 0 0 0 28 9 37 39 0 0 0 0 0 0 0 0 2 13 33 35 0 0 0 0 28 39 33 39 0 0 0 0 36 15 30 28 0 0 0 0 20 21 22 11
,
 0 0 34 1 0 0 0 0 40 40 39 40 0 0 0 0 23 23 1 0 0 0 0 0 37 38 7 0 0 0 0 0 0 0 0 0 1 0 38 38 0 0 0 0 0 1 3 0 0 0 0 0 0 13 40 0 0 0 0 0 28 28 0 40
,
 40 7 0 0 0 0 0 0 34 7 0 0 0 0 0 0 13 6 35 34 0 0 0 0 16 8 6 0 0 0 0 0 0 0 0 0 35 35 0 0 0 0 0 0 6 40 0 0 0 0 0 0 37 28 7 6 0 0 0 0 0 9 34 0
,
 24 40 0 0 0 0 0 0 3 17 0 0 0 0 0 0 17 0 20 17 0 0 0 0 38 1 15 21 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 28 9 0 0 0 0 0 0 36 35 19 28 0 0 0 0 40 6 31 22

`G:=sub<GL(8,GF(41))| [11,9,32,28,0,0,0,0,32,30,0,9,0,0,0,0,0,0,2,37,0,0,0,0,0,0,32,39,0,0,0,0,0,0,0,0,2,28,36,20,0,0,0,0,13,39,15,21,0,0,0,0,33,33,30,22,0,0,0,0,35,39,28,11],[0,40,23,37,0,0,0,0,0,40,23,38,0,0,0,0,34,39,1,7,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,13,28,0,0,0,0,38,3,40,0,0,0,0,0,38,0,0,40],[40,34,13,16,0,0,0,0,7,7,6,8,0,0,0,0,0,0,35,6,0,0,0,0,0,0,34,0,0,0,0,0,0,0,0,0,35,6,37,0,0,0,0,0,35,40,28,9,0,0,0,0,0,0,7,34,0,0,0,0,0,0,6,0],[24,3,17,38,0,0,0,0,40,17,0,1,0,0,0,0,0,0,20,15,0,0,0,0,0,0,17,21,0,0,0,0,0,0,0,0,32,28,36,40,0,0,0,0,0,9,35,6,0,0,0,0,0,0,19,31,0,0,0,0,0,0,28,22] >;`

C42.140D10 in GAP, Magma, Sage, TeX

`C_4^2._{140}D_{10}`
`% in TeX`

`G:=Group("C4^2.140D10");`
`// GroupNames label`

`G:=SmallGroup(320,1344);`
`// by ID`

`G=gap.SmallGroup(320,1344);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,219,184,1571,570,297,136,12550]);`
`// Polycyclic`

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

׿
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