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## G = C20.Q8order 160 = 25·5

### 2nd non-split extension by C20 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C5⋊2C8 — C20.Q8
 Lower central C5 — C10 — C20 — C20.Q8
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C20.Q8
G = < a,b,c | a20=b4=1, c2=a5b2, bab-1=a11, cac-1=a9, cbc-1=a15b-1 >

Smallest permutation representation of C20.Q8
Regular action 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)
(1 142 74 138)(2 153 75 129)(3 144 76 140)(4 155 77 131)(5 146 78 122)(6 157 79 133)(7 148 80 124)(8 159 61 135)(9 150 62 126)(10 141 63 137)(11 152 64 128)(12 143 65 139)(13 154 66 130)(14 145 67 121)(15 156 68 132)(16 147 69 123)(17 158 70 134)(18 149 71 125)(19 160 72 136)(20 151 73 127)(21 91 103 45)(22 82 104 56)(23 93 105 47)(24 84 106 58)(25 95 107 49)(26 86 108 60)(27 97 109 51)(28 88 110 42)(29 99 111 53)(30 90 112 44)(31 81 113 55)(32 92 114 46)(33 83 115 57)(34 94 116 48)(35 85 117 59)(36 96 118 50)(37 87 119 41)(38 98 120 52)(39 89 101 43)(40 100 102 54)
(1 50 79 81 11 60 69 91)(2 59 80 90 12 49 70 100)(3 48 61 99 13 58 71 89)(4 57 62 88 14 47 72 98)(5 46 63 97 15 56 73 87)(6 55 64 86 16 45 74 96)(7 44 65 95 17 54 75 85)(8 53 66 84 18 43 76 94)(9 42 67 93 19 52 77 83)(10 51 68 82 20 41 78 92)(21 157 108 138 31 147 118 128)(22 146 109 127 32 156 119 137)(23 155 110 136 33 145 120 126)(24 144 111 125 34 154 101 135)(25 153 112 134 35 143 102 124)(26 142 113 123 36 152 103 133)(27 151 114 132 37 141 104 122)(28 160 115 121 38 150 105 131)(29 149 116 130 39 159 106 140)(30 158 117 139 40 148 107 129)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 20 20 2 2 10 10 10 10 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + - + + + - - + image C1 C2 C2 C2 C4 Q8 D4 D5 SD16 D10 Dic10 C4×D5 C5⋊D4 D4.D5 Q8⋊D5 kernel C20.Q8 C2×C5⋊2C8 C4⋊Dic5 C5×C4⋊C4 C5⋊2C8 C20 C2×C10 C4⋊C4 C10 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 4 2 4 4 4 2 2

Matrix representation of C20.Q8 in GL4(𝔽41) generated by

 7 1 0 0 40 0 0 0 0 0 40 20 0 0 4 1
,
 32 0 0 0 0 32 0 0 0 0 9 0 0 0 5 32
,
 21 26 0 0 2 20 0 0 0 0 0 28 0 0 22 11
`G:=sub<GL(4,GF(41))| [7,40,0,0,1,0,0,0,0,0,40,4,0,0,20,1],[32,0,0,0,0,32,0,0,0,0,9,5,0,0,0,32],[21,2,0,0,26,20,0,0,0,0,0,22,0,0,28,11] >;`

C20.Q8 in GAP, Magma, Sage, TeX

`C_{20}.Q_8`
`% in TeX`

`G:=Group("C20.Q8");`
`// GroupNames label`

`G:=SmallGroup(160,15);`
`// by ID`

`G=gap.SmallGroup(160,15);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,313,31,297,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^4=1,c^2=a^5*b^2,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=a^15*b^-1>;`
`// generators/relations`

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