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## G = C42.141D10order 320 = 26·5

### 141st non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.141D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — C42.141D10
 Lower central C5 — C2×C10 — C42.141D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.141D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >

Subgroups: 926 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4, C2×C4 [×4], C2×C4 [×19], D4 [×6], Q8 [×10], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic5 [×2], Dic5 [×6], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4, C4.4D4, C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×8], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10 [×2], C23.38C23, C4×Dic5, C10.D4 [×6], C4⋊Dic5 [×4], D10⋊C4 [×2], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], D42D5 [×4], Q8×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C202Q8, C42⋊D5, Dic5.14D4 [×4], D10.12D4 [×4], C20.17D4, Dic5⋊Q8, C5×C4.4D4, C2×D42D5, C2×Q8×D5, C42.141D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- 1+4 [×2], C22×D5 [×7], C23.38C23, D4×D5 [×2], C23×D5, C2×D4×D5, D4.10D10 [×2], C42.141D10

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 D10 2- 1+4 D4×D5 D4.10D10 kernel C42.141D10 C20⋊2Q8 C42⋊D5 Dic5.14D4 D10.12D4 C20.17D4 Dic5⋊Q8 C5×C4.4D4 C2×D4⋊2D5 C2×Q8×D5 C4×D5 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 C10 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 4 2 2 8 2 2 2 4 8

Matrix representation of C42.141D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 14 0 34 0 0 0 0 14 0 34 0 0 34 0 27 0 0 0 0 34 0 27
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 30 32 0 0 0 0 9 11 0 0 0 0 0 0 30 32 0 0 0 0 9 11
,
 32 17 0 0 0 0 17 9 0 0 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 34 34 0 0 0 0 7 1 0 0
,
 9 24 0 0 0 0 24 32 0 0 0 0 0 0 0 0 7 7 0 0 0 0 40 34 0 0 34 34 0 0 0 0 1 7 0 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,34,0,0,0,0,14,0,34,0,0,34,0,27,0,0,0,0,34,0,27],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[32,17,0,0,0,0,17,9,0,0,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,7,34,0,0,0,0,7,40,0,0],[9,24,0,0,0,0,24,32,0,0,0,0,0,0,0,0,34,1,0,0,0,0,34,7,0,0,7,40,0,0,0,0,7,34,0,0] >;`

C42.141D10 in GAP, Magma, Sage, TeX

`C_4^2._{141}D_{10}`
`% in TeX`

`G:=Group("C4^2.141D10");`
`// GroupNames label`

`G:=SmallGroup(320,1347);`
`// by ID`

`G=gap.SmallGroup(320,1347);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,675,297,192,12550]);`
`// Polycyclic`

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

׿
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