D8xF5 - GroupNames

G = D₈×F₅ order 320 = 2⁶·5

Direct product of D₈ and F₅

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

Aliases: D₈×F₅, D₄₀⋊₅C₄, C₅⋊(C₄×D₈), C₈⋊₄(C₂×F₅), C₄₀⋊₂(C₂×C₄), D₄⋊D₅⋊₁C₄, (C₅×D₈)⋊₅C₄, D₄⋊₁(C₂×F₅), (C₈×F₅)⋊₁C₂, (D₄×F₅)⋊₁C₂, D₂₀⋊₁(C₂×C₄), D₅.D₈⋊₂C₂, (D₅×D₈).₅C₂, D₅.₂(C₂×D₈), C₂.₁₅(D₄×F₅), D₂₀⋊C₄⋊₁C₂, C₁₀.₁₄(C₄×D₄), (C₂×F₅).₁₀D₄, C₄⋊F₅.₁C₂², C₄.₁(C₂²×F₅), D₅.₂(C₄○D₈), D₁₀.₆₃(C₂×D₄), C₂₀.₁(C₂²×C₄), D₅⋊C₈.₉C₂², (D₄×D₅).₅C₂², (C₄×F₅).₉C₂², (C₄×D₅).₂₃C₂³, (C₈×D₅).₂₁C₂², Dic₅.₁(C₄○D₄), (C₅×D₄)⋊₁(C₂×C₄), C₅⋊₂C₈⋊₁₁(C₂×C₄), SmallGroup(320,1068)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — D₈×F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₄×F₅ — D₄×F₅ — D₈×F₅

Lower central

C₅ — C₁₀ — C₂₀ — D₈×F₅

Upper central

C₁ — C₂ — C₄ — D₈

Generators and relations for D₈×F₅
G = < a,b,c,d | a⁸=b²=c⁵=d⁴=1, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c³ >

Subgroups: 674 in 134 conjugacy classes, 44 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₈, C₈, C₂×C₄, D₄, D₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, D₈, D₈, C₂²×C₄, C₂×D₄, Dic₅, C₂₀, F₅, F₅, D₁₀, D₁₀, C₂×C₁₀, C₄×C₈, D₄⋊C₄, C₂.D₈, C₄×D₄, C₂×D₈, C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₄×D₅, D₂₀, C₅⋊D₄, C₅×D₄, C₂×F₅, C₂×F₅, C₂²×D₅, C₄×D₈, C₈×D₅, D₄₀, D₄⋊D₅, C₅×D₈, D₅⋊C₈, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, D₄×D₅, C₂²×F₅, C₈×F₅, D₅.D₈, D₂₀⋊C₄, D₅×D₈, D₄×F₅, D₈×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₈, C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₄×D₄, C₂×D₈, C₄○D₈, C₂×F₅, C₄×D₈, C₂²×F₅, D₄×F₅, D₈×F₅

Smallest permutation representation of D₈×F₅
►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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 22)(18 21)(19 20)(23 24)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)

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

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

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

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

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

35 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	5	8A	8B	8C	···	8H	10A	10B	10C	20	40A	40B
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	5	8	8	8	···	8	10	10	10	20	40	40
size	1	1	4	4	5	5	20	20	2	5	5	5	5	10	10	10	20	20	20	20	4	2	2	10	···	10	4	16	16	8	8	8

35 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	8	8
type	+	+	+	+	+	+				+		+		+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	C₄○D₄	D₈	C₄○D₈	F₅	C₂×F₅	C₂×F₅	D₄×F₅	D₈×F₅
kernel	D₈×F₅	C₈×F₅	D₅.D₈	D₂₀⋊C₄	D₅×D₈	D₄×F₅	D₄₀	D₄⋊D₅	C₅×D₈	C₂×F₅	Dic₅	F₅	D₅	D₈	C₈	D₄	C₂	C₁
# reps	1	1	1	2	1	2	2	4	2	2	2	4	4	1	1	2	1	2

Matrix representation of D₈×F₅ ►in GL₆(𝔽₄₁)

0	24	0	0	0	0
29	24	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	24	0	0	0	0
12	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

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

9	0	0	0	0	0
0	9	0	0	0	0
0	0	0	0	1	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(41))| [0,29,0,0,0,0,24,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,24,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

D₈×F₅ in GAP, Magma, Sage, TeX

D_8\times F_5

% in TeX

G:=Group("D8xF5");

// GroupNames label

G:=SmallGroup(320,1068);

// by ID

G=gap.SmallGroup(320,1068);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,851,438,102,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;

// generators/relations

G = D8×F5 order 320 = 26·5

Direct product of D8 and F5

G = D₈×F₅ order 320 = 2⁶·5

Direct product of D₈ and F₅