(C2xD4):F5 - GroupNames

G = (C₂×D₄)⋊F₅ order 320 = 2⁶·5

2^nd semidirect product of C₂×D₄ and F₅ acting via F₅/C₅=C₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×D₄)⋊₂F₅, (D₄×C₁₀)⋊₂C₄, (C₄×Dic₅)⋊₅C₄, C₅⋊₂(C₄²⋊C₄), C₂₀⋊D₄.₂C₂, C₂.₆(C₂³⋊F₅), D₁₀.D₄⋊₂C₂, (C₂²×D₅).₁₂D₄, C₁₀.₁₅(C₂³⋊C₄), (C₂×D₂₀).₄₀C₂², C₂².₁₉(C₂²⋊F₅), (C₂×C₄).₁(C₂×F₅), (C₂×C₂₀).₁₁(C₂×C₄), (C₂×C₁₀).₃₅(C₂²⋊C₄), SmallGroup(320,260)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — (C₂×D₄)⋊F₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×D₅ — C₂×D₂₀ — D₁₀.D₄ — (C₂×D₄)⋊F₅

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — (C₂×D₄)⋊F₅

Upper central

C₁ — C₂ — C₂² — C₂×C₄ — C₂×D₄

Generators and relations for (C₂×D₄)⋊F₅
G = < a,b,c,d,e | a²=b⁴=c²=d⁵=e⁴=1, ebe^-1=ab=ba, ac=ca, ad=da, eae^-1=ab², cbc=b^-1, bd=db, cd=dc, ece^-1=bc, ede^-1=d³ >

Subgroups: 586 in 86 conjugacy classes, 18 normal (14 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂×D₄, C₂×D₄, Dic₅, C₂₀, F₅, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₄⋊₁D₄, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₂×F₅, C₂²×D₅, C₂²×C₁₀, C₄²⋊C₄, C₄×Dic₅, C₂²⋊F₅, C₂×D₂₀, C₂×C₅⋊D₄, D₄×C₁₀, D₁₀.D₄, C₂₀⋊D₄, (C₂×D₄)⋊F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, F₅, C₂³⋊C₄, C₂×F₅, C₄²⋊C₄, C₂²⋊F₅, C₂³⋊F₅, (C₂×D₄)⋊F₅

Character table of (C₂×D₄)⋊F₅

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	5	10A	10B	10C	10D	10E	10F	10G	20A	20B
size	1	1	2	8	20	20	4	20	20	40	40	40	40	4	4	4	4	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	1	1	1	linear of order 2
ρ₃	1	1	1	-1	1	1	1	-1	-1	1	-1	-1	1	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₄	1	1	1	-1	1	1	1	-1	-1	-1	1	1	-1	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₅	1	1	1	1	-1	-1	1	-1	-1	i	-i	i	-i	1	1	1	1	1	1	1	1	1	1	linear of order 4
ρ₆	1	1	1	1	-1	-1	1	-1	-1	-i	i	-i	i	1	1	1	1	1	1	1	1	1	1	linear of order 4
ρ₇	1	1	1	-1	-1	-1	1	1	1	i	i	-i	-i	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 4
ρ₈	1	1	1	-1	-1	-1	1	1	1	-i	-i	i	i	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 4
ρ₉	2	2	2	0	2	-2	-2	0	0	0	0	0	0	2	2	2	2	0	0	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	2	0	-2	2	-2	0	0	0	0	0	0	2	2	2	2	0	0	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₁	4	-4	0	0	0	0	0	-2	2	0	0	0	0	4	0	-4	0	0	0	0	0	0	0	orthogonal lifted from C₄²⋊C₄
ρ₁₂	4	-4	0	0	0	0	0	2	-2	0	0	0	0	4	0	-4	0	0	0	0	0	0	0	orthogonal lifted from C₄²⋊C₄
ρ₁₃	4	4	4	4	0	0	4	0	0	0	0	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₁₄	4	4	-4	0	0	0	0	0	0	0	0	0	0	4	-4	4	-4	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₁₅	4	4	4	-4	0	0	4	0	0	0	0	0	0	-1	-1	-1	-1	1	1	1	1	-1	-1	orthogonal lifted from C₂×F₅
ρ₁₆	4	4	4	0	0	0	-4	0	0	0	0	0	0	-1	-1	-1	-1	√5	√5	-√5	-√5	1	1	orthogonal lifted from C₂²⋊F₅
ρ₁₇	4	4	4	0	0	0	-4	0	0	0	0	0	0	-1	-1	-1	-1	-√5	-√5	√5	√5	1	1	orthogonal lifted from C₂²⋊F₅
ρ₁₈	4	4	-4	0	0	0	0	0	0	0	0	0	0	-1	1	-1	1	2ζ₅⁴+2ζ₅³+1	2ζ₅²+2ζ₅+1	2ζ₅⁴+2ζ₅²+1	2ζ₅³+2ζ₅+1	√5	-√5	complex lifted from C₂³⋊F₅
ρ₁₉	4	4	-4	0	0	0	0	0	0	0	0	0	0	-1	1	-1	1	2ζ₅²+2ζ₅+1	2ζ₅⁴+2ζ₅³+1	2ζ₅³+2ζ₅+1	2ζ₅⁴+2ζ₅²+1	√5	-√5	complex lifted from C₂³⋊F₅
ρ₂₀	4	4	-4	0	0	0	0	0	0	0	0	0	0	-1	1	-1	1	2ζ₅³+2ζ₅+1	2ζ₅⁴+2ζ₅²+1	2ζ₅⁴+2ζ₅³+1	2ζ₅²+2ζ₅+1	-√5	√5	complex lifted from C₂³⋊F₅
ρ₂₁	4	4	-4	0	0	0	0	0	0	0	0	0	0	-1	1	-1	1	2ζ₅⁴+2ζ₅²+1	2ζ₅³+2ζ₅+1	2ζ₅²+2ζ₅+1	2ζ₅⁴+2ζ₅³+1	-√5	√5	complex lifted from C₂³⋊F₅
ρ₂₂	8	-8	0	0	0	0	0	0	0	0	0	0	0	-2	-2√5	2	2√5	0	0	0	0	0	0	orthogonal faithful
ρ₂₃	8	-8	0	0	0	0	0	0	0	0	0	0	0	-2	2√5	2	-2√5	0	0	0	0	0	0	orthogonal faithful

Smallest permutation representation of (C₂×D₄)⋊F₅
►On 40 points

Generators in S₄₀

(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

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

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

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

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

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

Matrix representation of (C₂×D₄)⋊F₅ ►in GL₈(𝔽₄₁)

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	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	36	0	40	0
0	0	0	0	39	29	0	40

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	0	0	0	0
0	0	0	0	40	25	0	0
0	0	0	0	36	1	0	0
0	0	0	0	5	29	40	5
0	0	0	0	31	16	16	1

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	1	0	0	0
0	0	0	0	5	40	0	0
0	0	0	0	36	11	1	36
0	0	0	0	9	0	0	40

40	40	0	0	0	0	0	0
36	35	0	0	0	0	0	0
39	39	6	40	0	0	0	0
0	0	1	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

0	3	1	1	0	0	0	0
21	18	6	40	0	0	0	0
25	25	20	0	0	0	0	0
0	10	3	3	0	0	0	0
0	0	0	0	40	0	16	0
0	0	0	0	0	30	39	5
0	0	0	0	0	1	1	0
0	0	0	0	0	17	37	11

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,36,39,0,0,0,0,0,1,0,29,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,36,5,31,0,0,0,0,25,1,29,16,0,0,0,0,0,0,40,16,0,0,0,0,0,0,5,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,5,36,9,0,0,0,0,0,40,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,40],[40,36,39,0,0,0,0,0,40,35,39,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,21,25,0,0,0,0,0,3,18,25,10,0,0,0,0,1,6,20,3,0,0,0,0,1,40,0,3,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,30,1,17,0,0,0,0,16,39,1,37,0,0,0,0,0,5,0,11] >;

(C₂×D₄)⋊F₅ in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes F_5

% in TeX

G:=Group("(C2xD4):F5");

// GroupNames label

G:=SmallGroup(320,260);

// by ID

G=gap.SmallGroup(320,260);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,297,136,1684,6278,3156]);

// Polycyclic

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

// generators/relations

Export

Character table of (C₂×D₄)⋊F₅ in TeX

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	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	36	0	40	0
0	0	0	0	39	29	0	40

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	0	0	0	0
0	0	0	0	40	25	0	0
0	0	0	0	36	1	0	0
0	0	0	0	5	29	40	5
0	0	0	0	31	16	16	1

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	1	0	0	0
0	0	0	0	5	40	0	0
0	0	0	0	36	11	1	36
0	0	0	0	9	0	0	40

40	40	0	0	0	0	0	0
36	35	0	0	0	0	0	0
39	39	6	40	0	0	0	0
0	0	1	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

0	3	1	1	0	0	0	0
21	18	6	40	0	0	0	0
25	25	20	0	0	0	0	0
0	10	3	3	0	0	0	0
0	0	0	0	40	0	16	0
0	0	0	0	0	30	39	5
0	0	0	0	0	1	1	0
0	0	0	0	0	17	37	11

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	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	36	0	40	0
0	0	0	0	39	29	0	40

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	0	0	0	0
0	0	0	0	40	25	0	0
0	0	0	0	36	1	0	0
0	0	0	0	5	29	40	5
0	0	0	0	31	16	16	1

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	1	0	0	0
0	0	0	0	5	40	0	0
0	0	0	0	36	11	1	36
0	0	0	0	9	0	0	40

40	40	0	0	0	0	0	0
36	35	0	0	0	0	0	0
39	39	6	40	0	0	0	0
0	0	1	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

0	3	1	1	0	0	0	0
21	18	6	40	0	0	0	0
25	25	20	0	0	0	0	0
0	10	3	3	0	0	0	0
0	0	0	0	40	0	16	0
0	0	0	0	0	30	39	5
0	0	0	0	0	1	1	0
0	0	0	0	0	17	37	11

G = (C2×D4)⋊F5 order 320 = 26·5

2nd semidirect product of C2×D4 and F5 acting via F5/C5=C4

G = (C₂×D₄)⋊F₅ order 320 = 2⁶·5

2^nd semidirect product of C₂×D₄ and F₅ acting via F₅/C₅=C₄

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	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	36	0	40	0
0	0	0	0	39	29	0	40

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	0	0	0	0
0	0	0	0	40	25	0	0
0	0	0	0	36	1	0	0
0	0	0	0	5	29	40	5
0	0	0	0	31	16	16	1

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	1	0	0	0
0	0	0	0	5	40	0	0
0	0	0	0	36	11	1	36
0	0	0	0	9	0	0	40

40	40	0	0	0	0	0	0
36	35	0	0	0	0	0	0
39	39	6	40	0	0	0	0
0	0	1	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

0	3	1	1	0	0	0	0
21	18	6	40	0	0	0	0
25	25	20	0	0	0	0	0
0	10	3	3	0	0	0	0
0	0	0	0	40	0	16	0
0	0	0	0	0	30	39	5
0	0	0	0	0	1	1	0
0	0	0	0	0	17	37	11