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## G = C20.6Q8order 160 = 25·5

### 3rd non-split extension by C20 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.6Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C10.D4 — C20.6Q8
 Lower central C5 — C2×C10 — C20.6Q8
 Upper central C1 — C22 — C42

Generators and relations for C20.6Q8
G = < a,b,c | a20=b4=1, c2=a10b2, ab=ba, cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 144 in 56 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, C10, C10, C42, C4⋊C4, Dic5, C20, C20, C2×C10, C42.C2, C2×Dic5, C2×C20, C2×C20, C10.D4, C4⋊Dic5, C4×C20, C20.6Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, D10, C42.C2, Dic10, C22×D5, C2×Dic10, C4○D20, C20.6Q8

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

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

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

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

46 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 5A 5B 10A ··· 10F 20A ··· 20X order 1 2 2 2 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 20 20 20 20 2 2 2 ··· 2 2 ··· 2

46 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + - + + - image C1 C2 C2 C2 Q8 D5 C4○D4 D10 Dic10 C4○D20 kernel C20.6Q8 C10.D4 C4⋊Dic5 C4×C20 C20 C42 C10 C2×C4 C4 C2 # reps 1 4 2 1 2 2 4 6 8 16

Matrix representation of C20.6Q8 in GL4(𝔽41) generated by

 25 30 0 0 27 39 0 0 0 0 18 0 0 0 15 16
,
 23 1 0 0 5 18 0 0 0 0 32 0 0 0 29 9
,
 39 18 0 0 2 2 0 0 0 0 15 39 0 0 31 26
`G:=sub<GL(4,GF(41))| [25,27,0,0,30,39,0,0,0,0,18,15,0,0,0,16],[23,5,0,0,1,18,0,0,0,0,32,29,0,0,0,9],[39,2,0,0,18,2,0,0,0,0,15,31,0,0,39,26] >;`

C20.6Q8 in GAP, Magma, Sage, TeX

`C_{20}._6Q_8`
`% in TeX`

`G:=Group("C20.6Q8");`
`// GroupNames label`

`G:=SmallGroup(160,91);`
`// by ID`

`G=gap.SmallGroup(160,91);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,55,218,86,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=b^4=1,c^2=a^10*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;`
`// generators/relations`

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