C2xA4xF5 - GroupNames

G = C₂×A₄×F₅ order 480 = 2⁵·3·5

Direct product of C₂, A₄ and F₅

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂×A₄×F₅, C₁₀⋊(C₄×A₄), D₅⋊(C₄×A₄), (C₂³×F₅)⋊C₃, C₂²⋊(C₆×F₅), (D₅×A₄)⋊₃C₄, (C₁₀×A₄)⋊₂C₄, (C₂²×F₅)⋊C₆, (C₂²×C₁₀)⋊C₁₂, (C₂²×D₅)⋊C₁₂, (C₂³×D₅).C₆, D₅.(C₂²×A₄), C₂³⋊₂(C₃×F₅), D₁₀.₄(C₂×A₄), (D₅×A₄).₃C₂², C₅⋊(C₂×C₄×A₄), (C₂×C₁₀)⋊(C₂×C₁₂), (C₂×D₅×A₄).₃C₂, (C₅×A₄)⋊₃(C₂×C₄), (C₂²×D₅).(C₂×C₆), SmallGroup(480,1192)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂×A₄×F₅

Chief series

C₁ — C₅ — C₂×C₁₀ — C₂²×D₅ — D₅×A₄ — A₄×F₅ — C₂×A₄×F₅

Lower central

C₂×C₁₀ — C₂×A₄×F₅

Upper central

C₁ — C₂

Generators and relations for C₂×A₄×F₅
G = < a,b,c,d,e,f | a²=b²=c²=d³=e⁵=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd^-1=bc=cb, be=eb, bf=fb, dcd^-1=b, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e³ >

Subgroups: 844 in 132 conjugacy classes, 30 normal (24 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₂×C₄, C₂³, C₂³, D₅, D₅, C₁₀, C₁₀, C₁₂, A₄, C₂×C₆, C₁₅, C₂²×C₄, C₂⁴, F₅, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₂, C₂×A₄, C₂×A₄, C₃×D₅, C₃₀, C₂³×C₄, C₂×F₅, C₂×F₅, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₄×A₄, C₂²×A₄, C₃×F₅, C₅×A₄, C₆×D₅, C₂²×F₅, C₂²×F₅, C₂³×D₅, C₂×C₄×A₄, D₅×A₄, C₆×F₅, C₁₀×A₄, C₂³×F₅, A₄×F₅, C₂×D₅×A₄, C₂×A₄×F₅
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, C₁₂, A₄, C₂×C₆, F₅, C₂×C₁₂, C₂×A₄, C₂×F₅, C₄×A₄, C₂²×A₄, C₃×F₅, C₂×C₄×A₄, C₆×F₅, A₄×F₅, C₂×A₄×F₅

Permutation representations of C₂×A₄×F₅
►On 30 points - transitive group 30T107

Generators in S₃₀

(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)

(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)

(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)

(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)

(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)

(1 6)(2 8 5 9)(3 10 4 7)(11 16)(12 18 15 19)(13 20 14 17)(21 26)(22 28 25 29)(23 30 24 27)

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

G:=Group( (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20), (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), (1,6)(2,8,5,9)(3,10,4,7)(11,16)(12,18,15,19)(13,20,14,17)(21,26)(22,28,25,29)(23,30,24,27) );

G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,6),(2,7),(3,8),(4,9),(5,10),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,21,11),(2,22,12),(3,23,13),(4,24,14),(5,25,15),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20)], [(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)], [(1,6),(2,8,5,9),(3,10,4,7),(11,16),(12,18,15,19),(13,20,14,17),(21,26),(22,28,25,29),(23,30,24,27)]])

G:=TransitiveGroup(30,107);

40 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	5	6A	6B	6C	6D	6E	6F	10A	10B	10C	12A	···	12H	15A	15B	30A	30B
order	1	2	2	2	2	2	2	2	3	3	4	4	4	4	4	4	4	4	5	6	6	6	6	6	6	10	10	10	12	···	12	15	15	30	30
size	1	1	3	3	5	5	15	15	4	4	5	5	5	5	15	15	15	15	4	4	4	20	20	20	20	4	12	12	20	···	20	16	16	16	16

40 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	12	12	3	3	3	3	3	4	4	4	4
type	+	+	+								+	+	+	+	+			+	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	A₄×F₅	C₂×A₄×F₅	A₄	C₂×A₄	C₂×A₄	C₄×A₄	C₄×A₄	F₅	C₂×F₅	C₃×F₅	C₆×F₅
kernel	C₂×A₄×F₅	A₄×F₅	C₂×D₅×A₄	C₂³×F₅	D₅×A₄	C₁₀×A₄	C₂²×F₅	C₂³×D₅	C₂²×D₅	C₂²×C₁₀	C₂	C₁	C₂×F₅	F₅	D₁₀	D₅	C₁₀	C₂×A₄	A₄	C₂³	C₂²
# reps	1	2	1	2	2	2	4	2	4	4	1	1	1	2	1	2	2	1	1	2	2

Matrix representation of C₂×A₄×F₅ ►in GL₇(𝔽₆₁)

60	0	0	0	0	0	0
0	60	0	0	0	0	0
0	0	60	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

60	0	0	0	0	0	0
0	60	0	0	0	0	0
14	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

60	0	0	0	0	0	0
48	1	0	0	0	0	0
0	0	60	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

13	59	0	0	0	0	0
0	48	1	0	0	0	0
0	14	0	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	60	60	60	60
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0

50	0	0	0	0	0	0
0	50	0	0	0	0	0
0	0	50	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0
0	0	0	60	60	60	60

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,0,14,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,48,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,59,48,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,1,0,0,0,0,60,0,0,1,0,0,0,60,0,0,0],[50,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,1,0,60] >;

C₂×A₄×F₅ in GAP, Magma, Sage, TeX

C_2\times A_4\times F_5

% in TeX

G:=Group("C2xA4xF5");

// GroupNames label

G:=SmallGroup(480,1192);

// by ID

G=gap.SmallGroup(480,1192);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,84,648,271,9414,1595]);

// Polycyclic

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

// generators/relations

G = C2×A4×F5 order 480 = 25·3·5

Direct product of C2, A4 and F5

G = C₂×A₄×F₅ order 480 = 2⁵·3·5

Direct product of C₂, A₄ and F₅