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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C8.C4
 Lower central C1 — C2 — C4 — C5×C8.C4
 Upper central C1 — C20 — C2×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 >

Smallest permutation representation of C5×C8.C4
On 80 points
Generators in S80
(1 41 31 37 9)(2 42 32 38 10)(3 43 25 39 11)(4 44 26 40 12)(5 45 27 33 13)(6 46 28 34 14)(7 47 29 35 15)(8 48 30 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 70 27 68 29 66 31 72)(26 69 28 67 30 65 32 71)(33 77 35 75 37 73 39 79)(34 76 36 74 38 80 40 78)

G:=sub<Sym(80)| (1,41,31,37,9)(2,42,32,38,10)(3,43,25,39,11)(4,44,26,40,12)(5,45,27,33,13)(6,46,28,34,14)(7,47,29,35,15)(8,48,30,36,16)(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)(25,70,27,68,29,66,31,72)(26,69,28,67,30,65,32,71)(33,77,35,75,37,73,39,79)(34,76,36,74,38,80,40,78)>;

G:=Group( (1,41,31,37,9)(2,42,32,38,10)(3,43,25,39,11)(4,44,26,40,12)(5,45,27,33,13)(6,46,28,34,14)(7,47,29,35,15)(8,48,30,36,16)(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)(25,70,27,68,29,66,31,72)(26,69,28,67,30,65,32,71)(33,77,35,75,37,73,39,79)(34,76,36,74,38,80,40,78) );

G=PermutationGroup([[(1,41,31,37,9),(2,42,32,38,10),(3,43,25,39,11),(4,44,26,40,12),(5,45,27,33,13),(6,46,28,34,14),(7,47,29,35,15),(8,48,30,36,16),(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),(25,70,27,68,29,66,31,72),(26,69,28,67,30,65,32,71),(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 2A 2B 4A 4B 4C 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 10G 10H 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P 40Q ··· 40AF order 1 2 2 4 4 4 5 5 5 5 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 40 ··· 40 size 1 1 2 1 1 2 1 1 1 1 2 2 2 2 4 4 4 4 1 1 1 1 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C5 C10 C10 C20 D4 Q8 C8.C4 C5×D4 C5×Q8 C5×C8.C4 kernel C5×C8.C4 C2×C40 C5×M4(2) C40 C8.C4 C2×C8 M4(2) C8 C20 C2×C10 C5 C4 C22 C1 # reps 1 1 2 4 4 4 8 16 1 1 4 4 4 16

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

 37 0 0 37
,
 14 0 0 3
,
 0 1 32 0
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

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