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## G = C42.282D10order 320 = 26·5

### 5th central extension by C42 of D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.282D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C42 — C42.282D10
 Lower central C5 — C10 — C42.282D10
 Upper central C1 — C42 — C4×C8

Generators and relations for C42.282D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c9 >

Subgroups: 350 in 118 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, D10, D10, C2×C10, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C52C8, C40, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C42.12C4, C2×C52C8, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C4×C52C8, C20.8Q8, D101C8, C4×C40, D5×C42, C42.282D10
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, M4(2), C22×C4, C4○D4, D10, C42⋊C2, C22×C8, C2×M4(2), C4×D5, C22×D5, C42.12C4, C8×D5, C8⋊D5, C2×C4×D5, C4○D20, C42⋊D5, D5×C2×C8, C2×C8⋊D5, C42.282D10

Smallest permutation representation of C42.282D10
On 160 points
Generators in S160
```(1 121 120 48)(2 122 81 49)(3 123 82 50)(4 124 83 51)(5 125 84 52)(6 126 85 53)(7 127 86 54)(8 128 87 55)(9 129 88 56)(10 130 89 57)(11 131 90 58)(12 132 91 59)(13 133 92 60)(14 134 93 61)(15 135 94 62)(16 136 95 63)(17 137 96 64)(18 138 97 65)(19 139 98 66)(20 140 99 67)(21 141 100 68)(22 142 101 69)(23 143 102 70)(24 144 103 71)(25 145 104 72)(26 146 105 73)(27 147 106 74)(28 148 107 75)(29 149 108 76)(30 150 109 77)(31 151 110 78)(32 152 111 79)(33 153 112 80)(34 154 113 41)(35 155 114 42)(36 156 115 43)(37 157 116 44)(38 158 117 45)(39 159 118 46)(40 160 119 47)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 90 99 21 30 110 119)(2 98 91 29 22 118 111 9)(3 28 92 117 23 8 112 97)(4 116 93 7 24 96 113 27)(5 6 94 95 25 26 114 115)(11 20 100 109 31 40 120 89)(12 108 101 39 32 88 81 19)(13 38 102 87 33 18 82 107)(14 86 103 17 34 106 83 37)(15 16 104 105 35 36 84 85)(41 147 124 44 61 127 144 64)(42 43 125 126 62 63 145 146)(45 143 128 80 65 123 148 60)(46 79 129 122 66 59 149 142)(47 121 130 58 67 141 150 78)(48 57 131 140 68 77 151 160)(49 139 132 76 69 159 152 56)(50 75 133 158 70 55 153 138)(51 157 134 54 71 137 154 74)(52 53 135 136 72 73 155 156)```

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10F 20A ··· 20X 40A ··· 40AF order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 1 ··· 1 10 ··· 10 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C8 D5 M4(2) C4○D4 D10 D10 C4×D5 C8×D5 C8⋊D5 C4○D20 kernel C42.282D10 C4×C5⋊2C8 C20.8Q8 D10⋊1C8 C4×C40 D5×C42 C4×Dic5 C2×C4×D5 C4×D5 C4×C8 C20 C20 C42 C2×C8 C2×C4 C4 C4 C4 # reps 1 1 2 2 1 1 4 4 16 2 4 4 2 4 8 16 16 16

Matrix representation of C42.282D10 in GL3(𝔽41) generated by

 40 0 0 0 32 0 0 0 32
,
 9 0 0 0 1 0 0 0 1
,
 38 0 0 0 6 15 0 3 21
,
 3 0 0 0 20 15 0 17 21
`G:=sub<GL(3,GF(41))| [40,0,0,0,32,0,0,0,32],[9,0,0,0,1,0,0,0,1],[38,0,0,0,6,3,0,15,21],[3,0,0,0,20,17,0,15,21] >;`

C42.282D10 in GAP, Magma, Sage, TeX

`C_4^2._{282}D_{10}`
`% in TeX`

`G:=Group("C4^2.282D10");`
`// GroupNames label`

`G:=SmallGroup(320,312);`
`// by ID`

`G=gap.SmallGroup(320,312);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,422,58,136,12550]);`
`// Polycyclic`

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

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