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G = C20⋊2Q8order 160 = 25·5

1st semidirect product of C20 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20⋊2Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C2×Dic10 — C20⋊2Q8
 Lower central C5 — C2×C10 — C20⋊2Q8
 Upper central C1 — C22 — C42

Generators and relations for C202Q8
G = < a,b,c | a20=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 192 in 68 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×Q8, Dic5, C20, C2×C10, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C4⋊Dic5, C4×C20, C2×Dic10, C202Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, Dic10, D20, C22×D5, C2×Dic10, C2×D20, C202Q8

Smallest permutation representation of C202Q8
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 31 50 67)(2 32 51 68)(3 33 52 69)(4 34 53 70)(5 35 54 71)(6 36 55 72)(7 37 56 73)(8 38 57 74)(9 39 58 75)(10 40 59 76)(11 21 60 77)(12 22 41 78)(13 23 42 79)(14 24 43 80)(15 25 44 61)(16 26 45 62)(17 27 46 63)(18 28 47 64)(19 29 48 65)(20 30 49 66)(81 143 134 114)(82 144 135 115)(83 145 136 116)(84 146 137 117)(85 147 138 118)(86 148 139 119)(87 149 140 120)(88 150 121 101)(89 151 122 102)(90 152 123 103)(91 153 124 104)(92 154 125 105)(93 155 126 106)(94 156 127 107)(95 157 128 108)(96 158 129 109)(97 159 130 110)(98 160 131 111)(99 141 132 112)(100 142 133 113)
(1 101 50 150)(2 120 51 149)(3 119 52 148)(4 118 53 147)(5 117 54 146)(6 116 55 145)(7 115 56 144)(8 114 57 143)(9 113 58 142)(10 112 59 141)(11 111 60 160)(12 110 41 159)(13 109 42 158)(14 108 43 157)(15 107 44 156)(16 106 45 155)(17 105 46 154)(18 104 47 153)(19 103 48 152)(20 102 49 151)(21 131 77 98)(22 130 78 97)(23 129 79 96)(24 128 80 95)(25 127 61 94)(26 126 62 93)(27 125 63 92)(28 124 64 91)(29 123 65 90)(30 122 66 89)(31 121 67 88)(32 140 68 87)(33 139 69 86)(34 138 70 85)(35 137 71 84)(36 136 72 83)(37 135 73 82)(38 134 74 81)(39 133 75 100)(40 132 76 99)```

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

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

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

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 D4 Q8 D5 D10 Dic10 D20 kernel C20⋊2Q8 C4⋊Dic5 C4×C20 C2×Dic10 C20 C20 C42 C2×C4 C4 C4 # reps 1 4 1 2 2 4 2 6 16 8

Matrix representation of C202Q8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 27 39 0 0 16 11
,
 1 2 0 0 40 40 0 0 0 0 1 0 0 0 0 1
,
 31 19 0 0 40 10 0 0 0 0 11 2 0 0 22 30
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,27,16,0,0,39,11],[1,40,0,0,2,40,0,0,0,0,1,0,0,0,0,1],[31,40,0,0,19,10,0,0,0,0,11,22,0,0,2,30] >;`

C202Q8 in GAP, Magma, Sage, TeX

`C_{20}\rtimes_2Q_8`
`% in TeX`

`G:=Group("C20:2Q8");`
`// GroupNames label`

`G:=SmallGroup(160,90);`
`// by ID`

`G=gap.SmallGroup(160,90);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,103,218,50,4613]);`
`// Polycyclic`

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

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