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G = D5×C4.D4  order 320 = 26·5

Direct product of D5 and C4.D4

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

Aliases: D5×C4.D4, M4(2)⋊14D10, (C4×D5).1D4, C4.146(D4×D5), C20.89(C2×D4), C23.6(C4×D5), C20.D4⋊3C2, (D5×M4(2))⋊5C2, (C2×C20).1C23, (C23×D5).2C4, (C2×D4).121D10, C20.46D4⋊5C2, C4.Dic5⋊1C22, (D4×C10).11C22, (C2×D20).42C22, D10.53(C22⋊C4), (C5×M4(2))⋊12C22, Dic5.21(C22⋊C4), (C2×D4×D5).2C2, C5⋊3(C2×C4.D4), (C2×C5⋊D4).1C4, (C2×C4×D5).5C22, C22.14(C2×C4×D5), (C5×C4.D4)⋊5C2, C2.13(D5×C22⋊C4), (C2×C4).1(C22×D5), C10.53(C2×C22⋊C4), (C22×C10).6(C2×C4), (C22×D5).2(C2×C4), (C2×Dic5).21(C2×C4), (C2×C10).109(C22×C4), SmallGroup(320,371)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — D5×C4.D4
Chief series
C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C2×D4×D5 — D5×C4.D4
Lower central
C5 — C10 — C2×C10 — D5×C4.D4
Upper central
C1 — C2 — C2×C4 — C4.D4

Generators and relations for D5×C4.D4
 G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 942 in 186 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C4.D4, C4.D4, C2×M4(2), C22×D4, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C2×C4.D4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C20.46D4, C20.D4, C5×C4.D4, D5×M4(2), C2×D4×D5, D5×C4.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4.D4, C2×C22⋊C4, C4×D5, C22×D5, C2×C4.D4, C2×C4×D5, D4×D5, D5×C22⋊C4, D5×C4.D4

Smallest permutation representation of D5×C4.D4
►On 40 points
Generators in S40
(1 31 11 33 22)(2 32 12 34 23)(3 25 13 35 24)(4 26 14 36 17)(5 27 15 37 18)(6 28 16 38 19)(7 29 9 39 20)(8 30 10 40 21)

(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D5A5B8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G10H20A20B20C20D40A···40H
order12222222224444558888888810101010101010102020202040···40
size1124455102020221010224444202020202244888844448···8

44 irreducible representations

dim1111111122222448
type+++++++++++++
imageC1C2C2C2C2C2C4C4D4D5D10D10C4×D5C4.D4D4×D5D5×C4.D4
kernelD5×C4.D4C20.46D4C20.D4C5×C4.D4D5×M4(2)C2×D4×D5C2×C5⋊D4C23×D5C4×D5C4.D4M4(2)C2×D4C23D5C4C1
# reps1211214442428242

Matrix representation of D5×C4.D4 ►in GL8(𝔽41)

400100000
040010000
330700000
033070000
00001000
00000100
00000010
00000001
,
10000000
01000000
804000000
080400000
000040000
000004000
000000400
000000040
,
400000000
040000000
004000000
000400000
00000100
000040000
000036001
000005400
,
01000000
10000000
00010000
00100000
0000537039
0000375390
000010364
000000436
,
040000000
10000000
000400000
00100000
0000375390
0000537039
000000436
000010364

G:=sub<GL(8,GF(41))| [40,0,33,0,0,0,0,0,0,40,0,33,0,0,0,0,1,0,7,0,0,0,0,0,0,1,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,8,0,0,0,0,0,0,1,0,8,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,36,0,0,0,0,0,1,0,0,5,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,37,1,0,0,0,0,0,37,5,0,0,0,0,0,0,0,39,36,4,0,0,0,0,39,0,4,36],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,37,5,0,1,0,0,0,0,5,37,0,0,0,0,0,0,39,0,4,36,0,0,0,0,0,39,36,4] >;
 

D5×C4.D4 in GAP, Magma, Sage, TeX 

D_5\times C_4.D_4
 
% in TeX
 
G:=Group("D5xC4.D4");
 
// GroupNames label
 
G:=SmallGroup(320,371);
 
// by ID
 
G=gap.SmallGroup(320,371);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,58,570,136,438,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^4=c^2,e^2=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^-1*d^3>;
 
// generators/relations
 

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