Copied to
clipboard

## G = C20.42C42order 320 = 26·5

### 5th non-split extension by C20 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C20.42C42
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — C23.21D10 — C20.42C42
 Lower central C5 — C10 — C20.42C42
 Upper central C1 — C2×C8 — C22×C8

Generators and relations for C20.42C42
G = < a,b,c | a20=b4=1, c4=a10, bab-1=a-1, ac=ca, bc=cb >

Subgroups: 286 in 130 conjugacy classes, 87 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C52C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C82M4(2), C2×C52C8, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C2×C40, C22×C20, C8×Dic5, C408C4, C2×C4.Dic5, C23.21D10, C22×C40, C20.42C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, Dic5, D10, C2×C42, C8○D4, C4×D5, C2×Dic5, C22×D5, C82M4(2), C4×Dic5, C2×C4×D5, C22×Dic5, D20.3C4, C2×C4×Dic5, C20.42C42

Smallest permutation representation of C20.42C42
On 160 points
Generators in S160
```(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)
(1 104 21 96)(2 103 22 95)(3 102 23 94)(4 101 24 93)(5 120 25 92)(6 119 26 91)(7 118 27 90)(8 117 28 89)(9 116 29 88)(10 115 30 87)(11 114 31 86)(12 113 32 85)(13 112 33 84)(14 111 34 83)(15 110 35 82)(16 109 36 81)(17 108 37 100)(18 107 38 99)(19 106 39 98)(20 105 40 97)(41 159 66 137)(42 158 67 136)(43 157 68 135)(44 156 69 134)(45 155 70 133)(46 154 71 132)(47 153 72 131)(48 152 73 130)(49 151 74 129)(50 150 75 128)(51 149 76 127)(52 148 77 126)(53 147 78 125)(54 146 79 124)(55 145 80 123)(56 144 61 122)(57 143 62 121)(58 142 63 140)(59 141 64 139)(60 160 65 138)
(1 56 6 41 11 46 16 51)(2 57 7 42 12 47 17 52)(3 58 8 43 13 48 18 53)(4 59 9 44 14 49 19 54)(5 60 10 45 15 50 20 55)(21 61 26 66 31 71 36 76)(22 62 27 67 32 72 37 77)(23 63 28 68 33 73 38 78)(24 64 29 69 34 74 39 79)(25 65 30 70 35 75 40 80)(81 127 96 122 91 137 86 132)(82 128 97 123 92 138 87 133)(83 129 98 124 93 139 88 134)(84 130 99 125 94 140 89 135)(85 131 100 126 95 121 90 136)(101 141 116 156 111 151 106 146)(102 142 117 157 112 152 107 147)(103 143 118 158 113 153 108 148)(104 144 119 159 114 154 109 149)(105 145 120 160 115 155 110 150)```

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 5A 5B 8A ··· 8H 8I 8J 8K 8L 8M ··· 8T 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 5 5 8 ··· 8 8 8 8 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 10 ··· 10 2 2 1 ··· 1 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 1 2 2 2 2 2 2 2 2 type + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D5 Dic5 D10 D10 C8○D4 C4×D5 C4×D5 D20.3C4 kernel C20.42C42 C8×Dic5 C40⋊8C4 C2×C4.Dic5 C23.21D10 C22×C40 C4.Dic5 C4⋊Dic5 C23.D5 C2×C40 C22×C8 C2×C8 C2×C8 C22×C4 C10 C2×C4 C23 C2 # reps 1 2 2 1 1 1 8 4 4 8 2 8 4 2 8 12 4 32

Matrix representation of C20.42C42 in GL4(𝔽41) generated by

 4 0 0 0 0 31 0 0 0 0 33 0 0 0 0 5
,
 0 1 0 0 40 0 0 0 0 0 0 1 0 0 40 0
,
 9 0 0 0 0 9 0 0 0 0 14 0 0 0 0 14
`G:=sub<GL(4,GF(41))| [4,0,0,0,0,31,0,0,0,0,33,0,0,0,0,5],[0,40,0,0,1,0,0,0,0,0,0,40,0,0,1,0],[9,0,0,0,0,9,0,0,0,0,14,0,0,0,0,14] >;`

C20.42C42 in GAP, Magma, Sage, TeX

`C_{20}._{42}C_4^2`
`% in TeX`

`G:=Group("C20.42C4^2");`
`// GroupNames label`

`G:=SmallGroup(320,728);`
`// by ID`

`G=gap.SmallGroup(320,728);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^4=1,c^4=a^10,b*a*b^-1=a^-1,a*c=c*a,b*c=c*b>;`
`// generators/relations`

׿
×
𝔽