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G = C5×C8.4Q8order 320 = 26·5

Direct product of C5 and C8.4Q8

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×C8.4Q8, C16.1C20, C80.10C4, C20.70D8, C40.20Q8, C8.5(C5×Q8), C4.19(C5×D8), (C2×C80).11C2, (C2×C16).5C10, C8.15(C2×C20), C20.89(C4⋊C4), (C2×C10).6Q16, C40.124(C2×C4), (C2×C20).410D4, C8.C4.3C10, C22.1(C5×Q16), C10.22(C2.D8), (C2×C40).431C22, C4.9(C5×C4⋊C4), C2.5(C5×C2.D8), (C2×C4).64(C5×D4), (C2×C8).89(C2×C10), (C5×C8.C4).6C2, SmallGroup(320,173)

Series: Derived Chief Lower central Upper central

C1C8 — C5×C8.4Q8
C1C2C4C2×C4C2×C8C2×C40C5×C8.C4 — C5×C8.4Q8
C1C2C4C8 — C5×C8.4Q8
C1C20C2×C20C2×C40 — C5×C8.4Q8

Generators and relations for C5×C8.4Q8
 G = < a,b,c,d | a5=b8=1, c4=b2, d2=bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b6c3 >

2C2
2C10
4C8
4C8
2M4(2)
2M4(2)
4C40
4C40
2C5×M4(2)
2C5×M4(2)

Smallest permutation representation of C5×C8.4Q8
On 160 points
Generators in S160
(1 112 40 160 90)(2 97 41 145 91)(3 98 42 146 92)(4 99 43 147 93)(5 100 44 148 94)(6 101 45 149 95)(7 102 46 150 96)(8 103 47 151 81)(9 104 48 152 82)(10 105 33 153 83)(11 106 34 154 84)(12 107 35 155 85)(13 108 36 156 86)(14 109 37 157 87)(15 110 38 158 88)(16 111 39 159 89)(17 78 64 113 139)(18 79 49 114 140)(19 80 50 115 141)(20 65 51 116 142)(21 66 52 117 143)(22 67 53 118 144)(23 68 54 119 129)(24 69 55 120 130)(25 70 56 121 131)(26 71 57 122 132)(27 72 58 123 133)(28 73 59 124 134)(29 74 60 125 135)(30 75 61 126 136)(31 76 62 127 137)(32 77 63 128 138)
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 44 38 48 42 36 46 40)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)(65 67 69 71 73 75 77 79)(66 68 70 72 74 76 78 80)(81 91 85 95 89 83 93 87)(82 92 86 96 90 84 94 88)(97 107 101 111 105 99 109 103)(98 108 102 112 106 100 110 104)(113 115 117 119 121 123 125 127)(114 116 118 120 122 124 126 128)(129 131 133 135 137 139 141 143)(130 132 134 136 138 140 142 144)(145 155 149 159 153 147 157 151)(146 156 150 160 154 148 158 152)
(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 26 13 30 9 18 5 22)(2 25 14 29 10 17 6 21)(3 24 15 28 11 32 7 20)(4 23 16 27 12 31 8 19)(33 64 45 52 41 56 37 60)(34 63 46 51 42 55 38 59)(35 62 47 50 43 54 39 58)(36 61 48 49 44 53 40 57)(65 98 69 110 73 106 77 102)(66 97 70 109 74 105 78 101)(67 112 71 108 75 104 79 100)(68 111 72 107 76 103 80 99)(81 141 93 129 89 133 85 137)(82 140 94 144 90 132 86 136)(83 139 95 143 91 131 87 135)(84 138 96 142 92 130 88 134)(113 149 117 145 121 157 125 153)(114 148 118 160 122 156 126 152)(115 147 119 159 123 155 127 151)(116 146 120 158 124 154 128 150)

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

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

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

110 conjugacy classes

class 1 2A2B4A4B4C5A5B5C5D8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G10H16A···16H20A···20H20I20J20K20L40A···40P40Q···40AF80A···80AF
order122444555588888888101010101010101016···1620···202020202040···4040···4080···80
size112112111122228888111122222···21···122222···28···82···2

110 irreducible representations

dim111111112222222222
type+++-++-
imageC1C2C2C4C5C10C10C20Q8D4D8Q16C5×Q8C5×D4C8.4Q8C5×D8C5×Q16C5×C8.4Q8
kernelC5×C8.4Q8C5×C8.C4C2×C80C80C8.4Q8C8.C4C2×C16C16C40C2×C20C20C2×C10C8C2×C4C5C4C22C1
# reps12144841611224488832

Matrix representation of C5×C8.4Q8 in GL2(𝔽241) generated by

980
098
,
300
08
,
1260
044
,
01
640
G:=sub<GL(2,GF(241))| [98,0,0,98],[30,0,0,8],[126,0,0,44],[0,64,1,0] >;

C5×C8.4Q8 in GAP, Magma, Sage, TeX

C_5\times C_8._4Q_8
% in TeX

G:=Group("C5xC8.4Q8");
// GroupNames label

G:=SmallGroup(320,173);
// by ID

G=gap.SmallGroup(320,173);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,708,2803,360,172,10085,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^6*c^3>;
// generators/relations

Export

Subgroup lattice of C5×C8.4Q8 in TeX

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