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G = C2×D4×F5  order 320 = 26·5

Direct product of C2, D4 and F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D4×F5, D10.14C24, D5⋊(C4×D4), C10⋊(C4×D4), C20⋊(C22×C4), (D4×D5)⋊10C4, D20⋊7(C2×C4), D10⋊(C22×C4), C4⋊F5⋊3C22, C23⋊5(C2×F5), C4⋊1(C22×F5), (D4×C10)⋊11C4, (C2×D20)⋊14C4, Dic5⋊(C22×C4), (C23×F5)⋊2C2, (C4×F5)⋊8C22, D10.69(C2×D4), C10.9(C23×C4), D5.2(C22×D4), C22⋊F5⋊4C22, (C2×F5).3C23, C2.10(C23×F5), C22⋊1(C22×F5), (D4×D5).16C22, (C4×D5).49C23, D10.50(C4○D4), (C22×F5)⋊3C22, (C23×D5).91C22, (C22×D5).151C23, C5⋊(C2×C4×D4), (C2×C4×F5)⋊5C2, (C2×C4⋊F5)⋊4C2, (C2×C4)⋊8(C2×F5), (C2×C20)⋊4(C2×C4), (C5×D4)⋊7(C2×C4), (C4×D5)⋊6(C2×C4), C5⋊D4⋊1(C2×C4), (C2×C5⋊D4)⋊6C4, (C2×C10)⋊(C22×C4), (C2×D4×D5).17C2, D5.2(C2×C4○D4), (C2×C22⋊F5)⋊7C2, (C22×C10)⋊4(C2×C4), (C2×Dic5)⋊17(C2×C4), (C22×D5)⋊12(C2×C4), (C2×C4×D5).216C22, SmallGroup(320,1595)

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, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, D4, C23, C23, D5, D5, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, F5, F5, D10, D10, D10, C2×C10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C2×F5, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C2×C4×D4, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C22×F5, C22×F5, C22×F5, C23×D5, C2×C4×F5, C2×C4⋊F5, D4×F5, C2×C22⋊F5, C2×D4×D5, C23×F5, C2×D4×F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, F5, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×F5, C2×C4×D4, C22×F5, D4×F5, 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 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C···4J4K···4X 5 10A10B10C10D10E10F10G20A20B
order1222222222222222444···44···45101010101010102020
size11112222555510101010225···510···104444888888

50 irreducible representations

dim111111111112244448
type+++++++++++++
imageC1C2C2C2C2C2C2C4C4C4C4D4C4○D4F5C2×F5C2×F5C2×F5D4×F5
kernelC2×D4×F5C2×C4×F5C2×C4⋊F5D4×F5C2×C22⋊F5C2×D4×D5C23×F5C2×D20D4×D5C2×C5⋊D4D4×C10C2×F5D10C2×D4C2×C4D4C23C2
# reps111821228424411422

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

-10000000
0-1000000
00100000
00010000
00001000
00000100
00000010
00000001
,
12000000
-1-1000000
00-1-20000
00110000
0000-1000
00000-100
000000-10
0000000-1
,
12000000
0-1000000
00-1-20000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
0000-1-1-1-1
00001000
00000100
00000010
,
10000000
01000000
00-100000
000-10000
00001000
00000001
00000100
0000-1-1-1-1

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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