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G = C42.200D10order 320 = 26·5

20th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.200D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C42 — C42.200D10
 Lower central C5 — C10 — C42.200D10
 Upper central C1 — C2×C4 — C4⋊C8

Generators and relations for C42.200D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2c9 >

Subgroups: 350 in 118 conjugacy classes, 61 normal (37 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, C20, D10, D10, C2×C10, C4×C8, C22⋊C8, C4⋊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, C203C8, C8×Dic5, D101C8, C5×C4⋊C8, D5×C42, C42.200D10
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, C2×C4×D5, D42D5, Q82D5, C4⋊C47D5, D5×C2×C8, D5×M4(2), C42.200D10

Smallest permutation representation of C42.200D10
On 160 points
Generators in S160
```(1 54 112 148)(2 149 113 55)(3 56 114 150)(4 151 115 57)(5 58 116 152)(6 153 117 59)(7 60 118 154)(8 155 119 61)(9 62 120 156)(10 157 81 63)(11 64 82 158)(12 159 83 65)(13 66 84 160)(14 121 85 67)(15 68 86 122)(16 123 87 69)(17 70 88 124)(18 125 89 71)(19 72 90 126)(20 127 91 73)(21 74 92 128)(22 129 93 75)(23 76 94 130)(24 131 95 77)(25 78 96 132)(26 133 97 79)(27 80 98 134)(28 135 99 41)(29 42 100 136)(30 137 101 43)(31 44 102 138)(32 139 103 45)(33 46 104 140)(34 141 105 47)(35 48 106 142)(36 143 107 49)(37 50 108 144)(38 145 109 51)(39 52 110 146)(40 147 111 53)
(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 82 91 21 30 102 111)(2 90 83 29 22 110 103 9)(3 28 84 109 23 8 104 89)(4 108 85 7 24 88 105 27)(5 6 86 87 25 26 106 107)(11 20 92 101 31 40 112 81)(12 100 93 39 32 120 113 19)(13 38 94 119 33 18 114 99)(14 118 95 17 34 98 115 37)(15 16 96 97 35 36 116 117)(41 160 145 76 61 140 125 56)(42 75 146 139 62 55 126 159)(43 138 147 54 63 158 127 74)(44 53 148 157 64 73 128 137)(45 156 149 72 65 136 129 52)(46 71 150 135 66 51 130 155)(47 134 151 50 67 154 131 70)(48 49 152 153 68 69 132 133)(57 144 121 60 77 124 141 80)(58 59 122 123 78 79 142 143)```

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10F 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 1 1 1 1 2 2 2 2 5 ··· 5 10 10 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 C8 D5 M4(2) C4○D4 D10 D10 C4×D5 C8×D5 D4⋊2D5 Q8⋊2D5 D5×M4(2) kernel C42.200D10 C20⋊3C8 C8×Dic5 D10⋊1C8 C5×C4⋊C8 D5×C42 C4×Dic5 C2×C4×D5 C4×D5 C4⋊C8 Dic5 C20 C42 C2×C8 C2×C4 C4 C4 C4 C2 # reps 1 1 2 2 1 1 4 4 16 2 4 4 2 4 8 16 2 2 4

Matrix representation of C42.200D10 in GL5(𝔽41)

 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 32 0 0 0 0 0 9
,
 32 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 27 0 0 0 0 0 6 6 0 0 0 35 1 0 0 0 0 0 0 1 0 0 0 40 0
,
 14 0 0 0 0 0 6 6 0 0 0 1 35 0 0 0 0 0 0 40 0 0 0 40 0

`G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,32,0,0,0,0,0,9],[32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[27,0,0,0,0,0,6,35,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,1,0],[14,0,0,0,0,0,6,1,0,0,0,6,35,0,0,0,0,0,0,40,0,0,0,40,0] >;`

C42.200D10 in GAP, Magma, Sage, TeX

`C_4^2._{200}D_{10}`
`% in TeX`

`G:=Group("C4^2.200D10");`
`// GroupNames label`

`G:=SmallGroup(320,460);`
`// by ID`

`G=gap.SmallGroup(320,460);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,422,219,58,136,12550]);`
`// Polycyclic`

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

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