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G = C8×D5order 80 = 24·5

Direct product of C8 and D5

Aliases: C8×D5, C403C2, C8Dic5, D10.4C4, C4.12D10, Dic5.4C4, C20.12C22, C53(C2×C8), C8(C52C8), C52C86C2, C2.1(C4×D5), C10.8(C2×C4), (C4×D5).7C2, SmallGroup(80,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C8×D5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5
 Lower central C5 — C8×D5
 Upper central C1 — C8

Generators and relations for C8×D5
G = < a,b,c | a8=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C8×D5
On 40 points
Generators in S40
(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)
(1 35 9 22 26)(2 36 10 23 27)(3 37 11 24 28)(4 38 12 17 29)(5 39 13 18 30)(6 40 14 19 31)(7 33 15 20 32)(8 34 16 21 25)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 13)(10 14)(11 15)(12 16)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)

G:=sub<Sym(40)| (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), (1,35,9,22,26)(2,36,10,23,27)(3,37,11,24,28)(4,38,12,17,29)(5,39,13,18,30)(6,40,14,19,31)(7,33,15,20,32)(8,34,16,21,25), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,13)(10,14)(11,15)(12,16)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)>;

G:=Group( (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), (1,35,9,22,26)(2,36,10,23,27)(3,37,11,24,28)(4,38,12,17,29)(5,39,13,18,30)(6,40,14,19,31)(7,33,15,20,32)(8,34,16,21,25), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,13)(10,14)(11,15)(12,16)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33) );

G=PermutationGroup([(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)], [(1,35,9,22,26),(2,36,10,23,27),(3,37,11,24,28),(4,38,12,17,29),(5,39,13,18,30),(6,40,14,19,31),(7,33,15,20,32),(8,34,16,21,25)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,13),(10,14),(11,15),(12,16),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,33)])

C8×D5 is a maximal subgroup of
C80⋊C2  D5⋊C16  C8.F5  C8⋊F5  C40⋊C4  D5.D8  C40.C4  D10.Q8  D20.3C4  D20.2C4  D83D5  SD163D5  Q8.D10  D152C8  C20.29D10  Dic5.4F5  C8.A5
C8×D5 is a maximal quotient of
C80⋊C2  C20.8Q8  D101C8  D152C8  C20.29D10  Dic5.4F5

32 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 10 20 20 20 20 40 ··· 40 size 1 1 5 5 1 1 5 5 2 2 1 1 1 1 5 5 5 5 2 2 2 2 2 2 2 ··· 2

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 D5 D10 C4×D5 C8×D5 kernel C8×D5 C5⋊2C8 C40 C4×D5 Dic5 D10 D5 C8 C4 C2 C1 # reps 1 1 1 1 2 2 8 2 2 4 8

Matrix representation of C8×D5 in GL2(𝔽41) generated by

 14 0 0 14
,
 0 38 14 34
,
 7 20 14 34
G:=sub<GL(2,GF(41))| [14,0,0,14],[0,14,38,34],[7,14,20,34] >;

C8×D5 in GAP, Magma, Sage, TeX

C_8\times D_5
% in TeX

G:=Group("C8xD5");
// GroupNames label

G:=SmallGroup(80,4);
// by ID

G=gap.SmallGroup(80,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,26,42,1604]);
// Polycyclic

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

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