Copied to
clipboard

G = D5xC4.D4  order 320 = 26·5

Direct product of D5 and C4.D4

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

Aliases: D5xC4.D4, M4(2):14D10, (C4xD5).1D4, C4.146(D4xD5), C20.89(C2xD4), C23.6(C4xD5), C20.D4:3C2, (D5xM4(2)):5C2, (C2xC20).1C23, (C23xD5).2C4, (C2xD4).121D10, C20.46D4:5C2, C4.Dic5:1C22, (D4xC10).11C22, (C2xD20).42C22, D10.53(C22:C4), (C5xM4(2)):12C22, Dic5.21(C22:C4), (C2xD4xD5).2C2, C5:3(C2xC4.D4), (C2xC5:D4).1C4, (C2xC4xD5).5C22, C22.14(C2xC4xD5), (C5xC4.D4):5C2, C2.13(D5xC22:C4), (C2xC4).1(C22xD5), C10.53(C2xC22:C4), (C22xC10).6(C2xC4), (C22xD5).2(C2xC4), (C2xDic5).21(C2xC4), (C2xC10).109(C22xC4), SmallGroup(320,371)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC10 — D5xC4.D4
Chief series
C1 — C5 — C10 — C20 — C2xC20 — C2xC4xD5 — C2xD4xD5 — D5xC4.D4
Lower central
C5 — C10 — C2xC10 — D5xC4.D4
Upper central
C1 — C2 — C2xC4 — C4.D4

Generators and relations for D5xC4.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, C2xC4, C2xC4, D4, C23, C23, D5, D5, C10, C10, C2xC8, M4(2), M4(2), C22xC4, C2xD4, C2xD4, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C4.D4, C4.D4, C2xM4(2), C22xD4, C5:2C8, C40, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C5xD4, C22xD5, C22xD5, C22xD5, C22xC10, C2xC4.D4, C8xD5, C8:D5, C4.Dic5, C5xM4(2), C2xC4xD5, C2xD20, D4xD5, C2xC5:D4, D4xC10, C23xD5, C20.46D4, C20.D4, C5xC4.D4, D5xM4(2), C2xD4xD5, D5xC4.D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22:C4, C22xC4, C2xD4, D10, C4.D4, C2xC22:C4, C4xD5, C22xD5, C2xC4.D4, C2xC4xD5, D4xD5, D5xC22:C4, D5xC4.D4

Smallest permutation representation of D5xC4.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+++++++++++++
imageC1C2C2C2C2C2C4C4D4D5D10D10C4xD5C4.D4D4xD5D5xC4.D4
kernelD5xC4.D4C20.46D4C20.D4C5xC4.D4D5xM4(2)C2xD4xD5C2xC5:D4C23xD5C4xD5C4.D4M4(2)C2xD4C23D5C4C1
# reps1211214442428242

Matrix representation of D5xC4.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] >;
 

D5xC4.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
 

׿
x
:
Z
F
o
wr
Q
<