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G = C5×C8.C4order 160 = 25·5

Direct product of C5 and C8.C4

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

Aliases: C5×C8.C4, C8.1C20, C40.10C4, C20.68D4, M4(2).2C10, (C2×C8).5C10, C4.8(C2×C20), C22.(C5×Q8), C4.19(C5×D4), (C2×C10).2Q8, (C2×C40).15C2, C20.66(C2×C4), C10.21(C4⋊C4), (C5×M4(2)).4C2, (C2×C20).119C22, C2.5(C5×C4⋊C4), (C2×C4).22(C2×C10), SmallGroup(160,58)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C8.C4
C1C2C4C2×C4C2×C20C5×M4(2) — C5×C8.C4
C1C2C4 — C5×C8.C4
C1C20C2×C20 — C5×C8.C4

Generators and relations for C5×C8.C4
 G = < a,b,c | a5=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C2
2C10
2C8
2C8
2C40
2C40

Smallest permutation representation of C5×C8.C4
On 80 points
Generators in S80
(1 41 27 37 9)(2 42 28 38 10)(3 43 29 39 11)(4 44 30 40 12)(5 45 31 33 13)(6 46 32 34 14)(7 47 25 35 15)(8 48 26 36 16)(17 67 76 55 64)(18 68 77 56 57)(19 69 78 49 58)(20 70 79 50 59)(21 71 80 51 60)(22 72 73 52 61)(23 65 74 53 62)(24 66 75 54 63)
(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)
(1 61 3 59 5 57 7 63)(2 60 4 58 6 64 8 62)(9 52 11 50 13 56 15 54)(10 51 12 49 14 55 16 53)(17 48 23 42 21 44 19 46)(18 47 24 41 22 43 20 45)(25 66 27 72 29 70 31 68)(26 65 28 71 30 69 32 67)(33 77 35 75 37 73 39 79)(34 76 36 74 38 80 40 78)

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

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

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

C5×C8.C4 is a maximal subgroup of
C40.7Q8  C40.6Q8  D40.6C4  C40.8D4  D40.5C4  M4(2).25D10  D4016C4  D4013C4  C8.20D20  C8.21D20  C8.24D20

70 conjugacy classes

class 1 2A2B4A4B4C5A5B5C5D8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G10H20A···20H20I20J20K20L40A···40P40Q···40AF
order122444555588888888101010101010101020···202020202040···4040···40
size112112111122224444111122221···122222···24···4

70 irreducible representations

dim11111111222222
type++++-
imageC1C2C2C4C5C10C10C20D4Q8C8.C4C5×D4C5×Q8C5×C8.C4
kernelC5×C8.C4C2×C40C5×M4(2)C40C8.C4C2×C8M4(2)C8C20C2×C10C5C4C22C1
# reps1124448161144416

Matrix representation of C5×C8.C4 in GL2(𝔽41) generated by

370
037
,
140
03
,
01
320
G:=sub<GL(2,GF(41))| [37,0,0,37],[14,0,0,3],[0,32,1,0] >;

C5×C8.C4 in GAP, Magma, Sage, TeX

C_5\times C_8.C_4
% in TeX

G:=Group("C5xC8.C4");
// GroupNames label

G:=SmallGroup(160,58);
// by ID

G=gap.SmallGroup(160,58);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,127,2403,117,88]);
// Polycyclic

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

Export

Subgroup lattice of C5×C8.C4 in TeX

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