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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C8.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C40 — C5×M5(2) — C5×C8.C8
 Lower central C1 — C2 — C4 — C5×C8.C8
 Upper central C1 — C40 — C2×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 >

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 2A 2B 4A 4B 4C ··· 4G 5A 5B 5C 5D 8A 8B 8C 8D 8E ··· 8J 10A 10B 10C 10D 10E 10F 10G 10H 16A ··· 16H 20A ··· 20H 20I ··· 20AB 40A ··· 40P 40Q ··· 40AN 80A ··· 80AF order 1 2 2 4 4 4 ··· 4 5 5 5 5 8 8 8 8 8 ··· 8 10 10 10 10 10 10 10 10 16 ··· 16 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 80 ··· 80 size 1 1 2 1 1 2 ··· 2 1 1 1 1 1 1 1 1 2 ··· 2 1 1 1 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 D4 Q8 M4(2) C5×D4 C5×Q8 C8.C8 C5×M4(2) C5×C8.C8 kernel C5×C8.C8 C4×C40 C5×M5(2) C4×C20 C2×C40 C8.C8 C40 C4×C8 M5(2) C42 C2×C8 C8 C40 C40 C2×C10 C8 C8 C5 C22 C1 # reps 1 1 2 2 2 4 8 4 8 8 8 32 1 1 2 4 4 8 8 32

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

 16 0 0 16
,
 14 0 0 38
,
 0 26 31 0
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

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