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## G = Q8×C20order 160 = 25·5

### Direct product of C20 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×C20
G = < a,b,c | a20=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 70 conjugacy classes, 64 normal (16 characteristic)
C1, C2 [×3], C4 [×8], C4 [×3], C22, C5, C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4 [×3], C2×Q8, C20 [×8], C20 [×3], C2×C10, C4×Q8, C2×C20, C2×C20 [×6], C5×Q8 [×4], C4×C20 [×3], C5×C4⋊C4 [×3], Q8×C10, Q8×C20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], Q8 [×2], C23, C10 [×7], C22×C4, C2×Q8, C4○D4, C20 [×4], C2×C10 [×7], C4×Q8, C2×C20 [×6], C5×Q8 [×2], C22×C10, C22×C20, Q8×C10, C5×C4○D4, Q8×C20

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

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4P 5A 5B 5C 5D 10A ··· 10L 20A ··· 20P 20Q ··· 20BL order 1 2 2 2 4 4 4 4 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 1 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 1 ··· 1 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 Q8 C4○D4 C5×Q8 C5×C4○D4 kernel Q8×C20 C4×C20 C5×C4⋊C4 Q8×C10 C5×Q8 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C20 C10 C4 C2 # reps 1 3 3 1 8 4 12 12 4 32 2 2 8 8

Matrix representation of Q8×C20 in GL3(𝔽41) generated by

 32 0 0 0 2 0 0 0 2
,
 40 0 0 0 12 40 0 22 29
,
 40 0 0 0 16 34 0 25 25
G:=sub<GL(3,GF(41))| [32,0,0,0,2,0,0,0,2],[40,0,0,0,12,22,0,40,29],[40,0,0,0,16,25,0,34,25] >;

Q8×C20 in GAP, Magma, Sage, TeX

Q_8\times C_{20}
% in TeX

G:=Group("Q8xC20");
// GroupNames label

G:=SmallGroup(160,180);
// by ID

G=gap.SmallGroup(160,180);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,247,554]);
// Polycyclic

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

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