Copied to
clipboard

## G = D20.6C8order 320 = 26·5

### The non-split extension by D20 of C8 acting through Inn(D20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D20.6C8
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D20.3C4 — D20.6C8
 Lower central C5 — C10 — D20.6C8
 Upper central C1 — C16 — C2×C16

Generators and relations for D20.6C8
G = < a,b,c | a20=b2=1, c8=a10, bab=a-1, ac=ca, bc=cb >

Subgroups: 214 in 84 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, C16 [×2], C16 [×2], C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C2×C16, C2×C16 [×2], M5(2) [×3], C8○D4, C52C8 [×2], C40 [×2], Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, D4○C16, C52C16 [×2], C80 [×2], C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C2×C40, C4○D20, D5×C16 [×2], C80⋊C2 [×2], C20.4C8, C2×C80, D20.3C4, D20.6C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], C22×C4, D10 [×3], C22×C8, C4×D5 [×2], C22×D5, D4○C16, C8×D5 [×2], C2×C4×D5, D5×C2×C8, D20.6C8

Smallest permutation representation of D20.6C8
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 140)(135 139)(136 138)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)
(1 150 56 116 88 140 26 65 11 160 46 106 98 130 36 75)(2 151 57 117 89 121 27 66 12 141 47 107 99 131 37 76)(3 152 58 118 90 122 28 67 13 142 48 108 100 132 38 77)(4 153 59 119 91 123 29 68 14 143 49 109 81 133 39 78)(5 154 60 120 92 124 30 69 15 144 50 110 82 134 40 79)(6 155 41 101 93 125 31 70 16 145 51 111 83 135 21 80)(7 156 42 102 94 126 32 71 17 146 52 112 84 136 22 61)(8 157 43 103 95 127 33 72 18 147 53 113 85 137 23 62)(9 158 44 104 96 128 34 73 19 148 54 114 86 138 24 63)(10 159 45 105 97 129 35 74 20 149 55 115 87 139 25 64)```

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A ··· 10F 16A ··· 16H 16I 16J 16K 16L 16M ··· 16T 20A ··· 20H 40A ··· 40P 80A ··· 80AF order 1 2 2 2 2 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 8 8 10 ··· 10 16 ··· 16 16 16 16 16 16 ··· 16 20 ··· 20 40 ··· 40 80 ··· 80 size 1 1 2 10 10 1 1 2 10 10 2 2 1 1 1 1 2 2 10 10 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 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D5 D10 D10 C4×D5 C4×D5 D4○C16 C8×D5 C8×D5 D20.6C8 kernel D20.6C8 D5×C16 C80⋊C2 C20.4C8 C2×C80 D20.3C4 C8⋊D5 C4.Dic5 C4○D20 Dic10 D20 C5⋊D4 C2×C16 C16 C2×C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 4 4 8 2 4 2 4 4 8 8 8 32

Matrix representation of D20.6C8 in GL2(𝔽241) generated by

 156 200 41 119
,
 0 240 240 0
,
 44 0 0 44
`G:=sub<GL(2,GF(241))| [156,41,200,119],[0,240,240,0],[44,0,0,44] >;`

D20.6C8 in GAP, Magma, Sage, TeX

`D_{20}._6C_8`
`% in TeX`

`G:=Group("D20.6C8");`
`// GroupNames label`

`G:=SmallGroup(320,528);`
`// by ID`

`G=gap.SmallGroup(320,528);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,58,80,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^2=1,c^8=a^10,b*a*b=a^-1,a*c=c*a,b*c=c*b>;`
`// generators/relations`

׿
×
𝔽