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G = C5×C8.C8order 320 = 26·5

Direct product of C5 and C8.C8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C8.C8, C8.1C40, C40.13C8, C40.22Q8, C40.109D4, C42.8C20, M5(2).4C10, C8.6(C5×Q8), (C2×C8).9C20, C4.8(C2×C40), C8.29(C5×D4), C20.82(C2×C8), (C4×C40).28C2, (C4×C20).40C4, (C4×C8).10C10, (C2×C40).52C4, C10.24(C4⋊C8), C20.99(C4⋊C4), (C5×M5(2)).8C2, (C2×C40).443C22, (C2×C10).35M4(2), C22.5(C5×M4(2)), C2.5(C5×C4⋊C8), C4.19(C5×C4⋊C4), (C2×C4).68(C2×C20), (C2×C8).97(C2×C10), (C2×C20).502(C2×C4), SmallGroup(320,169)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C8.C8
C1C2C4C8C2×C8C2×C40C5×M5(2) — C5×C8.C8
C1C2C4 — C5×C8.C8
C1C40C2×C40 — C5×C8.C8

Generators and relations for C5×C8.C8
 G = < a,b,c | a5=b8=1, c8=b4, ab=ba, ac=ca, cbc-1=b3 >

2C2
2C4
2C4
2C10
2C2×C4
2C20
2C20
2C16
2C16
2C2×C20
2C80
2C80

Smallest permutation representation of C5×C8.C8
On 80 points
Generators in S80
(1 78 17 62 43)(2 79 18 63 44)(3 80 19 64 45)(4 65 20 49 46)(5 66 21 50 47)(6 67 22 51 48)(7 68 23 52 33)(8 69 24 53 34)(9 70 25 54 35)(10 71 26 55 36)(11 72 27 56 37)(12 73 28 57 38)(13 74 29 58 39)(14 75 30 59 40)(15 76 31 60 41)(16 77 32 61 42)
(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)(17 23 29 19 25 31 21 27)(18 20 22 24 26 28 30 32)(33 39 45 35 41 47 37 43)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 56 62 52 58 64 54 60)(65 67 69 71 73 75 77 79)(66 72 78 68 74 80 70 76)
(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)

G:=sub<Sym(80)| (1,78,17,62,43)(2,79,18,63,44)(3,80,19,64,45)(4,65,20,49,46)(5,66,21,50,47)(6,67,22,51,48)(7,68,23,52,33)(8,69,24,53,34)(9,70,25,54,35)(10,71,26,55,36)(11,72,27,56,37)(12,73,28,57,38)(13,74,29,58,39)(14,75,30,59,40)(15,76,31,60,41)(16,77,32,61,42), (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16)(17,23,29,19,25,31,21,27)(18,20,22,24,26,28,30,32)(33,39,45,35,41,47,37,43)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,56,62,52,58,64,54,60)(65,67,69,71,73,75,77,79)(66,72,78,68,74,80,70,76), (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)>;

G:=Group( (1,78,17,62,43)(2,79,18,63,44)(3,80,19,64,45)(4,65,20,49,46)(5,66,21,50,47)(6,67,22,51,48)(7,68,23,52,33)(8,69,24,53,34)(9,70,25,54,35)(10,71,26,55,36)(11,72,27,56,37)(12,73,28,57,38)(13,74,29,58,39)(14,75,30,59,40)(15,76,31,60,41)(16,77,32,61,42), (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16)(17,23,29,19,25,31,21,27)(18,20,22,24,26,28,30,32)(33,39,45,35,41,47,37,43)(34,36,38,40,42,44,46,48)(49,51,53,55,57,59,61,63)(50,56,62,52,58,64,54,60)(65,67,69,71,73,75,77,79)(66,72,78,68,74,80,70,76), (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) );

G=PermutationGroup([(1,78,17,62,43),(2,79,18,63,44),(3,80,19,64,45),(4,65,20,49,46),(5,66,21,50,47),(6,67,22,51,48),(7,68,23,52,33),(8,69,24,53,34),(9,70,25,54,35),(10,71,26,55,36),(11,72,27,56,37),(12,73,28,57,38),(13,74,29,58,39),(14,75,30,59,40),(15,76,31,60,41),(16,77,32,61,42)], [(1,7,13,3,9,15,5,11),(2,4,6,8,10,12,14,16),(17,23,29,19,25,31,21,27),(18,20,22,24,26,28,30,32),(33,39,45,35,41,47,37,43),(34,36,38,40,42,44,46,48),(49,51,53,55,57,59,61,63),(50,56,62,52,58,64,54,60),(65,67,69,71,73,75,77,79),(66,72,78,68,74,80,70,76)], [(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)])

140 conjugacy classes

class 1 2A2B4A4B4C···4G5A5B5C5D8A8B8C8D8E···8J10A10B10C10D10E10F10G10H16A···16H20A···20H20I···20AB40A···40P40Q···40AN80A···80AF
order122444···4555588888···8101010101010101016···1620···2020···2040···4040···4080···80
size112112···2111111112···2111122224···41···12···21···12···24···4

140 irreducible representations

dim11111111111122222222
type++++-
imageC1C2C2C4C4C5C8C10C10C20C20C40D4Q8M4(2)C5×D4C5×Q8C8.C8C5×M4(2)C5×C8.C8
kernelC5×C8.C8C4×C40C5×M5(2)C4×C20C2×C40C8.C8C40C4×C8M5(2)C42C2×C8C8C40C40C2×C10C8C8C5C22C1
# reps1122248488832112448832

Matrix representation of C5×C8.C8 in GL2(𝔽41) generated by

160
016
,
140
038
,
026
310
G:=sub<GL(2,GF(41))| [16,0,0,16],[14,0,0,38],[0,31,26,0] >;

C5×C8.C8 in GAP, Magma, Sage, TeX

C_5\times C_8.C_8
% in TeX

G:=Group("C5xC8.C8");
// GroupNames label

G:=SmallGroup(320,169);
// by ID

G=gap.SmallGroup(320,169);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,148,5043,248,102,124]);
// Polycyclic

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

Export

Subgroup lattice of C5×C8.C8 in TeX

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