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## G = C20⋊Q8order 160 = 25·5

### The semidirect product of C20 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20⋊Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C20⋊Q8
 Lower central C5 — C2×C10 — C20⋊Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C20⋊Q8
G = < a,b,c | a20=b4=1, c2=b2, bab-1=a11, cac-1=a9, cbc-1=b-1 >

Subgroups: 192 in 68 conjugacy classes, 37 normal (19 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×Dic10, C20⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, Dic10, C22×D5, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8

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

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - - + + - + - image C1 C2 C2 C2 C2 C2 D4 Q8 Q8 D5 D10 Dic10 D4×D5 Q8×D5 kernel C20⋊Q8 C4×Dic5 C10.D4 C4⋊Dic5 C5×C4⋊C4 C2×Dic10 Dic5 Dic5 C20 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 1 2 1 1 2 2 2 2 2 6 8 2 2

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

 6 2 0 0 2 35 0 0 0 0 7 1 0 0 40 0
,
 0 1 0 0 40 0 0 0 0 0 11 9 0 0 32 30
,
 35 39 0 0 39 6 0 0 0 0 9 22 0 0 0 32
`G:=sub<GL(4,GF(41))| [6,2,0,0,2,35,0,0,0,0,7,40,0,0,1,0],[0,40,0,0,1,0,0,0,0,0,11,32,0,0,9,30],[35,39,0,0,39,6,0,0,0,0,9,0,0,0,22,32] >;`

C20⋊Q8 in GAP, Magma, Sage, TeX

`C_{20}\rtimes Q_8`
`% in TeX`

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

`G:=SmallGroup(160,109);`
`// by ID`

`G=gap.SmallGroup(160,109);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,218,188,50,4613]);`
`// Polycyclic`

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

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