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## G = C42.143D10order 320 = 26·5

### 143rd non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.143D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C42.143D10
 Lower central C5 — C2×C10 — C42.143D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C42.143D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >

Subgroups: 854 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×6], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4, C4.4D4 [×3], C41D4, Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10 [×2], C22.53C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C4×Dic10, C4×D20, Dic54D4 [×2], D10.12D4 [×2], Dic5.5D4 [×2], C22.D20 [×2], D4×Dic5, C20⋊D4, Q8×Dic5, C20.23D4, C5×C4.4D4, C42.143D10
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], C22.53C24, D42D5 [×2], C23×D5, C2×D42D5, D5×C4○D4, D48D10, C42.143D10

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

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H ··· 4O 4P 4Q 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 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 20 20 2 2 2 2 4 4 4 10 ··· 10 20 20 2 2 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

53 irreducible representations

 dim 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 D5 C4○D4 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊2D5 D5×C4○D4 D4⋊8D10 kernel C42.143D10 C4×Dic10 C4×D20 Dic5⋊4D4 D10.12D4 Dic5.5D4 C22.D20 D4×Dic5 C20⋊D4 Q8×Dic5 C20.23D4 C5×C4.4D4 C4.4D4 Dic5 C20 C42 C22⋊C4 C2×D4 C2×Q8 C10 C4 C2 C2 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 4 4 2 8 2 2 1 4 4 4

Matrix representation of C42.143D10 in GL6(𝔽41)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 16 32
,
 32 16 0 0 0 0 36 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 16 32
,
 40 20 0 0 0 0 0 1 0 0 0 0 0 0 6 6 0 0 0 0 35 1 0 0 0 0 0 0 2 8 0 0 0 0 15 39
,
 40 20 0 0 0 0 4 1 0 0 0 0 0 0 6 6 0 0 0 0 1 35 0 0 0 0 0 0 39 33 0 0 0 0 16 2

`G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,16,0,0,0,0,0,32],[32,36,0,0,0,0,16,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,0,32],[40,0,0,0,0,0,20,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,2,15,0,0,0,0,8,39],[40,4,0,0,0,0,20,1,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,0,0,0,0,39,16,0,0,0,0,33,2] >;`

C42.143D10 in GAP, Magma, Sage, TeX

`C_4^2._{143}D_{10}`
`% in TeX`

`G:=Group("C4^2.143D10");`
`// GroupNames label`

`G:=SmallGroup(320,1353);`
`// by ID`

`G=gap.SmallGroup(320,1353);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,297,80,12550]);`
`// 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*b^2,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

׿
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