D5:C4wrC2 - GroupNames

G = D₅:C₄wrC₂ order 320 = 2⁶·5

The semidirect product of D₅ and C₄wrC₂ acting via C₄wrC₂/C₄oD₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₅:C₄wrC₂, (D₄xD₅):₈C₄, C₄oD₄:₁F₅, D₄:₅(C₂xF₅), Q₈:₅(C₂xF₅), (Q₈xD₅):₈C₄, C₄oD₂₀:₁C₄, D₂₀:₅(C₂xC₄), D₄:F₅:₇C₂, Q₈:₂F₅:₇C₂, (C₄xF₅):₇C₂², C₄.F₅:₅C₂², Dic₁₀:₅(C₂xC₄), D₁₀.₁₀(C₂xD₄), (C₄xD₅).₁₂₁D₄, D₅:M₄(2):₅C₂, C₄.₂₁(C₂²xF₅), C₂₀.₂₁(C₂²xC₄), (C₄xD₅).₄₃C₂³, (C₂²xD₅).₇₁D₄, C₄.₄₆(C₂²:F₅), C₂₀.₄₆(C₂²:C₄), (C₂xDic₅).₁₂₃D₄, Dic₅.₁₁₄(C₂xD₄), C₂².₄(C₂²:F₅), D₁₀.₄₈(C₂²:C₄), D₄:₂D₅.₁₃C₂², Q₈:₂D₅.₁₃C₂², Dic₅.₁₅(C₂²:C₄), C₅:₃(C₂xC₄wrC₂), (C₂xC₄xF₅):₄C₂, (C₅xC₄oD₄):₁C₄, (C₅xD₄):₅(C₂xC₄), (C₅xQ₈):₅(C₂xC₄), (D₅xC₄oD₄).₅C₂, (C₂xC₄).₈₉(C₂xF₅), (C₂xC₂₀).₆₆(C₂xC₄), (C₄xD₅).₂₇(C₂xC₄), C₂.₃₄(C₂xC₂²:F₅), C₁₀.₃₃(C₂xC₂²:C₄), (C₂xC₄xD₅).₂₁₀C₂², (C₂xC₁₀).₄(C₂²:C₄), SmallGroup(320,1130)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₅:C₄wrC₂

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄xD₅ — C₄xF₅ — C₂xC₄xF₅ — D₅:C₄wrC₂

Lower central

C₅ — C₁₀ — C₂₀ — D₅:C₄wrC₂

Upper central

C₁ — C₄ — C₂xC₄ — C₄oD₄

Generators and relations for D₅:C₄wrC₂
G = < a,b,c,d,e | a⁵=b²=c⁴=d²=e⁴=1, bab=a^-1, ac=ca, ad=da, eae^-1=a³, bc=cb, bd=db, ebe^-1=a²b, dcd=c^-1, ce=ec, ede^-1=c^-1d >

Subgroups: 730 in 170 conjugacy classes, 50 normal (40 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₂xC₈, M₄(2), C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₄oD₄, Dic₅, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₄wrC₂, C₂xC₄², C₂xM₄(2), C₂xC₄oD₄, C₅:C₈, Dic₁₀, Dic₁₀, C₄xD₅, C₄xD₅, D₂₀, D₂₀, C₂xDic₅, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₂xF₅, C₂²xD₅, C₂²xD₅, C₂xC₄wrC₂, D₅:C₈, C₄.F₅, C₄xF₅, C₄xF₅, C₂².F₅, C₂xC₄xD₅, C₂xC₄xD₅, C₄oD₂₀, C₄oD₂₀, D₄xD₅, D₄xD₅, D₄:₂D₅, D₄:₂D₅, Q₈xD₅, Q₈:₂D₅, C₅xC₄oD₄, C₂²xF₅, D₄:F₅, Q₈:₂F₅, D₅:M₄(2), C₂xC₄xF₅, D₅xC₄oD₄, D₅:C₄wrC₂
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²:C₄, C₂²xC₄, C₂xD₄, F₅, C₄wrC₂, C₂xC₂²:C₄, C₂xF₅, C₂xC₄wrC₂, C₂²:F₅, C₂²xF₅, C₂xC₂²:F₅, D₅:C₄wrC₂

Smallest permutation representation of D₅:C₄wrC₂
►On 40 points

Generators in S₄₀

(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)

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

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

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

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	···	4O	4P	5	8A	8B	8C	8D	10A	10B	10C	10D	20A	20B	20C	20D	20E
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	···	4	4	5	8	8	8	8	10	10	10	10	20	20	20	20	20
size	1	1	2	4	5	5	10	20	1	1	2	4	5	5	10	···	10	20	4	20	20	20	20	4	8	8	8	4	4	8	8	8

38 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4	4	4	8
type	+	+	+	+	+	+					+	+	+		+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₄	D₄	C₄wrC₂	F₅	C₂xF₅	C₂xF₅	C₂xF₅	C₂²:F₅	C₂²:F₅	D₅:C₄wrC₂
kernel	D₅:C₄wrC₂	D₄:F₅	Q₈:₂F₅	D₅:M₄(2)	C₂xC₄xF₅	D₅xC₄oD₄	C₄oD₂₀	D₄xD₅	Q₈xD₅	C₅xC₄oD₄	C₄xD₅	C₂xDic₅	C₂²xD₅	D₅	C₄oD₄	C₂xC₄	D₄	Q₈	C₄	C₂²	C₁
# reps	1	2	2	1	1	1	2	2	2	2	2	1	1	8	1	1	1	1	2	2	2

Matrix representation of D₅:C₄wrC₂ ►in GL₆(F₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	40	40	40	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	40	40	40	40
0	0	0	0	0	1

32	32	0	0	0	0
0	9	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	9	0	0	0	0
23	32	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

40	36	0	0	0	0
0	9	0	0	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	40	40	40	40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,40,1],[32,0,0,0,0,0,32,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,23,0,0,0,0,9,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,36,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

D₅:C₄wrC₂ in GAP, Magma, Sage, TeX

D_5\rtimes C_4\wr C_2

% in TeX

G:=Group("D5:C4wrC2");

// GroupNames label

G:=SmallGroup(320,1130);

// by ID

G=gap.SmallGroup(320,1130);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,438,102,6278,1595]);

// Polycyclic

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

// generators/relations

G = D5:C4wrC2 order 320 = 26·5

The semidirect product of D5 and C4wrC2 acting via C4wrC2/C4oD4=C2

G = D₅:C₄wrC₂ order 320 = 2⁶·5

The semidirect product of D₅ and C₄wrC₂ acting via C₄wrC₂/C₄oD₄=C₂