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G = D5:(C4.D4)  order 320 = 26·5

The semidirect product of D5 and C4.D4 acting via C4.D4/C2xD4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5:(C4.D4), (C4xD5).38D4, (C2xD4).10F5, C23.F5:4C2, C23.4(C2xF5), (C2xD20).13C4, (D4xC10).11C4, D5:M4(2):3C2, Dic5.6(C2xD4), (C23xD5).5C4, C4.15(C22:F5), C20.15(C22:C4), C22.F5:1C22, D10.44(C22:C4), C22.14(C22xF5), (C2xDic5).174C23, (C2xD4xD5).15C2, C5:1(C2xC4.D4), (C2xC4).38(C2xF5), (C2xC20).57(C2xC4), C2.23(C2xC22:F5), C10.22(C2xC22:C4), (C22xD5).9(C2xC4), (C2xC4xD5).202C22, (C22xC10).29(C2xC4), (C2xC10).81(C22xC4), (C2xC5:D4).90C22, SmallGroup(320,1116)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC10 — D5:(C4.D4)
Chief series
C1 — C5 — C10 — Dic5 — C2xDic5 — C22.F5 — D5:M4(2) — D5:(C4.D4)
Lower central
C5 — C10 — C2xC10 — D5:(C4.D4)
Upper central
C1 — C2 — C2xC4 — C2xD4

Generators and relations for D5:(C4.D4)
 G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=c, bab=cac-1=a-1, dad-1=eae-1=a3, cbc-1=a3b, dbd-1=ebe-1=a2b, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 970 in 186 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, C23, C23, D5, D5, C10, C10, C2xC8, M4(2), C22xC4, C2xD4, C2xD4, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C4.D4, C2xM4(2), C22xD4, C5:C8, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C5xD4, C22xD5, C22xD5, C22xD5, C22xC10, C2xC4.D4, D5:C8, C4.F5, C22.F5, C2xC4xD5, C2xD20, D4xD5, C2xC5:D4, D4xC10, C23xD5, C23.F5, D5:M4(2), C2xD4xD5, D5:(C4.D4)
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, F5, C4.D4, C2xC22:C4, C2xF5, C2xC4.D4, C22:F5, C22xF5, C2xC22:F5, D5:(C4.D4)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D 5 8A···8H10A10B10C10D10E10F10G20A20B
order1222222222444458···8101010101010102020
size1124455102020221010420···20444888888

32 irreducible representations

dim11111112444448
type+++++++++++
imageC1C2C2C2C4C4C4D4F5C4.D4C2xF5C2xF5C22:F5D5:(C4.D4)
kernelD5:(C4.D4)C23.F5D5:M4(2)C2xD4xD5C2xD20D4xC10C23xD5C4xD5C2xD4D5C2xC4C23C4C1
# reps14212244121242

Matrix representation of D5:(C4.D4) ►in GL8(Z)

-1-1-1-10000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
-1-1-1-10000
00010000
00100000
01000000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
11110000
000-10000
00-100000
00000100
0000-1000
00000001
000000-10
,
10110000
11010000
-10-100000
-100-10000
00000010
0000000-1
00000100
00001000
,
10110000
11010000
-10-100000
-100-10000
00000010
00000001
00000100
0000-1000

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

D5:(C4.D4) in GAP, Magma, Sage, TeX 

D_5\rtimes (C_4.D_4)
 
% in TeX
 
G:=Group("D5:(C4.D4)");
 
// GroupNames label
 
G:=SmallGroup(320,1116);
 
// by ID
 
G=gap.SmallGroup(320,1116);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,297,136,1684,6278,1595]);
 
// 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=c*a*c^-1=a^-1,d*a*d^-1=e*a*e^-1=a^3,c*b*c^-1=a^3*b,d*b*d^-1=e*b*e^-1=a^2*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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