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## G = C2×D4×F5order 320 = 26·5

### Direct product of C2, D4 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D4×F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — C23×F5 — C2×D4×F5
 Lower central C5 — C10 — C2×D4×F5
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4×F5
G = < a,b,c,d,e | a2=b4=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 1594 in 426 conjugacy classes, 156 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×2], C4 [×12], C22, C22 [×4], C22 [×34], C5, C2×C4, C2×C4 [×39], D4 [×4], D4 [×12], C23 [×2], C23 [×19], D5 [×2], D5 [×2], D5 [×4], C10, C10 [×2], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×21], C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×2], C20 [×2], F5 [×4], F5 [×6], D10 [×2], D10 [×8], D10 [×20], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C2×F5 [×12], C2×F5 [×22], C22×D5, C22×D5 [×10], C22×D5 [×8], C22×C10 [×2], C2×C4×D4, C4×F5 [×4], C4⋊F5 [×4], C22⋊F5 [×8], C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C22×F5 [×2], C22×F5 [×10], C22×F5 [×8], C23×D5 [×2], C2×C4×F5, C2×C4⋊F5, D4×F5 [×8], C2×C22⋊F5 [×2], C2×D4×D5, C23×F5 [×2], C2×D4×F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, F5, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×F5 [×7], C2×C4×D4, C22×F5 [×7], D4×F5 [×2], C23×F5, C2×D4×F5

Smallest permutation representation of C2×D4×F5
On 40 points
Generators in S40
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 36 6 31)(2 37 7 32)(3 38 8 33)(4 39 9 34)(5 40 10 35)(11 21 16 26)(12 22 17 27)(13 23 18 28)(14 24 19 29)(15 25 20 30)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(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 26)(2 28 5 29)(3 30 4 27)(6 21)(7 23 10 24)(8 25 9 22)(11 36)(12 38 15 39)(13 40 14 37)(16 31)(17 33 20 34)(18 35 19 32)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C ··· 4J 4K ··· 4X 5 10A 10B 10C 10D 10E 10F 10G 20A 20B order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 ··· 4 4 ··· 4 5 10 10 10 10 10 10 10 20 20 size 1 1 1 1 2 2 2 2 5 5 5 5 10 10 10 10 2 2 5 ··· 5 10 ··· 10 4 4 4 4 8 8 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 F5 C2×F5 C2×F5 C2×F5 D4×F5 kernel C2×D4×F5 C2×C4×F5 C2×C4⋊F5 D4×F5 C2×C22⋊F5 C2×D4×D5 C23×F5 C2×D20 D4×D5 C2×C5⋊D4 D4×C10 C2×F5 D10 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 1 8 2 1 2 2 8 4 2 4 4 1 1 4 2 2

Matrix representation of C2×D4×F5 in GL8(ℤ)

 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 2 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 -2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 1 2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 -1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,-1,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1] >;

C2×D4×F5 in GAP, Magma, Sage, TeX

C_2\times D_4\times F_5
% in TeX

G:=Group("C2xD4xF5");
// GroupNames label

G:=SmallGroup(320,1595);
// by ID

G=gap.SmallGroup(320,1595);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,6278,818]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

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