D5xC2^3:C4 - GroupNames

G = D₅×C₂³⋊C₄ order 320 = 2⁶·5

Direct product of D₅ and C₂³⋊C₄

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

Aliases: D₅×C₂³⋊C₄, (C₂×D₂₀)⋊₆C₄, C₂³⋊₁(C₄×D₅), (C₂³×D₅)⋊₃C₄, C₂²⋊C₄⋊₂₂D₁₀, C₂³⋊Dic₅⋊₄C₂, C₂².₂₆(D₄×D₅), (C₂×D₄).₁₂₀D₁₀, (C₂²×D₅).₁₈D₄, C₂³.D₅⋊₃C₂², C₂³.₄(C₂²×D₅), (D₄×C₁₀).₁₀C₂², C₂³.₁D₁₀⋊₄C₂, (C₂²×C₁₀).₄C₂³, D₁₀.₅₂(C₂²⋊C₄), (C₂³×D₅).₁₀C₂², (C₂×C₄×D₅)⋊₂C₄, (C₂×C₄)⋊₁(C₄×D₅), C₅⋊₄(C₂×C₂³⋊C₄), (C₂×D₄×D₅).₁C₂, (C₂×C₂₀)⋊₁(C₂×C₄), (C₂×C₅⋊D₄)⋊₂C₄, (C₅×C₂³⋊C₄)⋊₆C₂, C₂².₁₃(C₂×C₄×D₅), (C₂²×C₁₀)⋊₁(C₂×C₄), (D₅×C₂²⋊C₄)⋊₂₂C₂, (C₂×Dic₅)⋊₁(C₂×C₄), (C₂×C₁₀).₁₉(C₂×D₄), (C₂²×D₅)⋊₁(C₂×C₄), C₂.₁₂(D₅×C₂²⋊C₄), C₁₀.₅₂(C₂×C₂²⋊C₄), (C₂×C₅⋊D₄).₄C₂², (C₅×C₂²⋊C₄)⋊₃₃C₂², (C₂×C₁₀).₁₀₈(C₂²×C₄), SmallGroup(320,370)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — D₅×C₂³⋊C₄

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂³×D₅ — C₂×D₄×D₅ — D₅×C₂³⋊C₄

Lower central

C₅ — C₁₀ — C₂×C₁₀ — D₅×C₂³⋊C₄

Upper central

C₁ — C₂ — C₂³ — C₂³⋊C₄

Generators and relations for D₅×C₂³⋊C₄
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e²=f⁴=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf^-1=cde, fdf^-1=de=ed, ef=fe >

Subgroups: 1102 in 210 conjugacy classes, 53 normal (31 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₅, D₅, C₁₀, C₁₀, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂²×D₄, C₄×D₅, D₂₀, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂×C₂³⋊C₄, D₁₀⋊C₄, C₂³.D₅, C₅×C₂²⋊C₄, C₂×C₄×D₅, C₂×C₄×D₅, C₂×D₂₀, D₄×D₅, C₂×C₅⋊D₄, D₄×C₁₀, C₂³×D₅, C₂³.₁D₁₀, C₂³⋊Dic₅, C₅×C₂³⋊C₄, D₅×C₂²⋊C₄, C₂×D₄×D₅, D₅×C₂³⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₅, C₂²⋊C₄, C₂²×C₄, C₂×D₄, D₁₀, C₂³⋊C₄, C₂×C₂²⋊C₄, C₄×D₅, C₂²×D₅, C₂×C₂³⋊C₄, C₂×C₄×D₅, D₄×D₅, D₅×C₂²⋊C₄, D₅×C₂³⋊C₄

Smallest permutation representation of D₅×C₂³⋊C₄
►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 24)(2 25)(3 21)(4 22)(5 23)(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 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)

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

(11 16)(12 17)(13 18)(14 19)(15 20)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	4A	···	4E	4F	···	4J	5A	5B	10A	10B	10C	···	10H	10I	10J	20A	···	20J
order	1	2	2	2	2	2	2	2	2	2	2	2	4	···	4	4	···	4	5	5	10	10	10	···	10	10	10	20	···	20
size	1	1	2	2	2	4	5	5	10	10	10	20	4	···	4	20	···	20	2	2	2	2	4	···	4	8	8	8	···	8

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	8
type	+	+	+	+	+	+					+	+	+	+			+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₅	D₁₀	D₁₀	C₄×D₅	C₄×D₅	C₂³⋊C₄	D₄×D₅	D₅×C₂³⋊C₄
kernel	D₅×C₂³⋊C₄	C₂³.₁D₁₀	C₂³⋊Dic₅	C₅×C₂³⋊C₄	D₅×C₂²⋊C₄	C₂×D₄×D₅	C₂×C₄×D₅	C₂×D₂₀	C₂×C₅⋊D₄	C₂³×D₅	C₂²×D₅	C₂³⋊C₄	C₂²⋊C₄	C₂×D₄	C₂×C₄	C₂³	D₅	C₂²	C₁
# reps	1	2	1	1	2	1	2	2	2	2	4	2	4	2	4	4	2	4	2

Matrix representation of D₅×C₂³⋊C₄ ►in GL₆(𝔽₄₁)

0	1	0	0	0	0
40	34	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

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

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

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	40	0	0
0	0	1	1	1	32
0	0	23	0	23	40

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,40,0,0,0,0,1,40,0,0,0,0,0,9,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,9,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,23,0,0,0,40,1,0,0,0,0,0,1,23,0,0,0,0,32,40] >;

D₅×C₂³⋊C₄ in GAP, Magma, Sage, TeX

D_5\times C_2^3\rtimes C_4

% in TeX

G:=Group("D5xC2^3:C4");

// GroupNames label

G:=SmallGroup(320,370);

// by ID

G=gap.SmallGroup(320,370);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,58,570,438,12550]);

// Polycyclic

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

// generators/relations

G = D5×C23⋊C4 order 320 = 26·5

Direct product of D5 and C23⋊C4

G = D₅×C₂³⋊C₄ order 320 = 2⁶·5

Direct product of D₅ and C₂³⋊C₄