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## G = C42.202D10order 320 = 26·5

### 22nd non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.202D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C42 — C42.202D10
 Lower central C5 — C2×C10 — C42.202D10
 Upper central C1 — C2×C4 — C4⋊C8

Generators and relations for C42.202D10
G = < a,b,c,d | a4=b4=1, c10=b, d2=a2b-1, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c9 >

Subgroups: 350 in 110 conjugacy classes, 53 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, C20, C20, C20, D10, D10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C42.6C4, C2×C52C8, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C203C8, C408C4, D101C8, C5×C4⋊C8, D5×C42, C42.202D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, M4(2), C22×C4, C4○D4, D10, C42⋊C2, C2×M4(2), C4×D5, C22×D5, C42.6C4, C8⋊D5, C2×C4×D5, D42D5, Q82D5, C4⋊C47D5, C2×C8⋊D5, D5×M4(2), C42.202D10

Smallest permutation representation of C42.202D10
On 160 points
Generators in S160
```(1 139 62 87)(2 88 63 140)(3 141 64 89)(4 90 65 142)(5 143 66 91)(6 92 67 144)(7 145 68 93)(8 94 69 146)(9 147 70 95)(10 96 71 148)(11 149 72 97)(12 98 73 150)(13 151 74 99)(14 100 75 152)(15 153 76 101)(16 102 77 154)(17 155 78 103)(18 104 79 156)(19 157 80 105)(20 106 41 158)(21 159 42 107)(22 108 43 160)(23 121 44 109)(24 110 45 122)(25 123 46 111)(26 112 47 124)(27 125 48 113)(28 114 49 126)(29 127 50 115)(30 116 51 128)(31 129 52 117)(32 118 53 130)(33 131 54 119)(34 120 55 132)(35 133 56 81)(36 82 57 134)(37 135 58 83)(38 84 59 136)(39 137 60 85)(40 86 61 138)
(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 52 61 21 30 72 41)(2 60 53 29 22 80 73 9)(3 28 54 79 23 8 74 59)(4 78 55 7 24 58 75 27)(5 6 56 57 25 26 76 77)(11 20 62 71 31 40 42 51)(12 70 63 39 32 50 43 19)(13 38 64 49 33 18 44 69)(14 48 65 17 34 68 45 37)(15 16 66 67 35 36 46 47)(81 82 123 124 101 102 143 144)(83 100 125 142 103 120 145 122)(84 141 126 119 104 121 146 99)(85 118 127 160 105 98 147 140)(86 159 128 97 106 139 148 117)(87 96 129 138 107 116 149 158)(88 137 130 115 108 157 150 95)(89 114 131 156 109 94 151 136)(90 155 132 93 110 135 152 113)(91 92 133 134 111 112 153 154)```

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 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 5 5 8 8 8 8 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 10 ··· 10 2 2 4 4 4 4 20 20 20 20 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 D5 M4(2) C4○D4 M4(2) D10 D10 C4×D5 C8⋊D5 D4⋊2D5 Q8⋊2D5 D5×M4(2) kernel C42.202D10 C20⋊3C8 C40⋊8C4 D10⋊1C8 C5×C4⋊C8 D5×C42 C4×Dic5 C2×C4×D5 C4⋊C8 C20 C20 D10 C42 C2×C8 C2×C4 C4 C4 C4 C2 # reps 1 1 2 2 1 1 4 4 2 4 4 4 2 4 8 16 2 2 4

Matrix representation of C42.202D10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 39 32
,
 32 0 0 0 0 32 0 0 0 0 40 0 0 0 0 40
,
 1 40 0 0 1 8 0 0 0 0 9 40 0 0 0 32
,
 40 1 0 0 8 1 0 0 0 0 32 1 0 0 2 9
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,39,0,0,0,32],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[1,1,0,0,40,8,0,0,0,0,9,0,0,0,40,32],[40,8,0,0,1,1,0,0,0,0,32,2,0,0,1,9] >;`

C42.202D10 in GAP, Magma, Sage, TeX

`C_4^2._{202}D_{10}`
`% in TeX`

`G:=Group("C4^2.202D10");`
`// GroupNames label`

`G:=SmallGroup(320,462);`
`// by ID`

`G=gap.SmallGroup(320,462);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,1094,219,58,136,12550]);`
`// Polycyclic`

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

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