F5xSL(2,3) - GroupNames

G = F₅×SL₂(𝔽₃) order 480 = 2⁵·3·5

Direct product of F₅ and SL₂(𝔽₃)

Aliases: F₅×SL₂(𝔽₃), (Q₈×F₅)⋊C₃, Q₈⋊(C₃×F₅), (C₅×Q₈)⋊C₁₂, (C₂×F₅).A₄, (Q₈×D₅).C₆, C₂.₃(A₄×F₅), C₁₀.₂(C₄×A₄), C₅⋊(C₄×SL₂(𝔽₃)), D₅.(C₄.A₄), D₁₀.₃(C₂×A₄), D₅.(C₂×SL₂(𝔽₃)), (C₅×SL₂(𝔽₃))⋊₃C₄, (D₅×SL₂(𝔽₃)).₃C₂, SmallGroup(480,965)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₅×Q₈ — F₅×SL₂(𝔽₃)

Chief series

C₁ — C₂ — C₁₀ — C₅×Q₈ — Q₈×D₅ — D₅×SL₂(𝔽₃) — F₅×SL₂(𝔽₃)

Lower central

C₅×Q₈ — F₅×SL₂(𝔽₃)

Upper central

C₁ — C₂

Generators and relations for F₅×SL₂(𝔽₃)
G = < a,b,c,d,e | a⁵=b⁴=c⁴=e³=1, d²=c², bab^-1=a³, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd^-1=c^-1, ece^-1=d, ede^-1=cd >

Subgroups: 378 in 60 conjugacy classes, 18 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₅, C₆, C₂×C₄, Q₈, Q₈, D₅, C₁₀, C₁₂, C₂×C₆, C₁₅, C₄², C₄⋊C₄, C₂×Q₈, Dic₅, C₂₀, F₅, F₅, D₁₀, SL₂(𝔽₃), C₂×C₁₂, C₃×D₅, C₃₀, C₄×Q₈, Dic₁₀, C₄×D₅, C₅×Q₈, C₂×F₅, C₂×F₅, C₂×SL₂(𝔽₃), C₃×F₅, C₆×D₅, C₄×F₅, C₄⋊F₅, Q₈×D₅, C₄×SL₂(𝔽₃), C₅×SL₂(𝔽₃), C₆×F₅, Q₈×F₅, D₅×SL₂(𝔽₃), F₅×SL₂(𝔽₃)
Quotients: C₁, C₂, C₃, C₄, C₆, C₁₂, A₄, F₅, SL₂(𝔽₃), C₂×A₄, C₄×A₄, C₂×SL₂(𝔽₃), C₄.A₄, C₃×F₅, C₄×SL₂(𝔽₃), A₄×F₅, F₅×SL₂(𝔽₃)

Smallest permutation representation of F₅×SL₂(𝔽₃)
►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 13)(2 15 5 11)(3 12 4 14)(6 39 10 37)(7 36 9 40)(8 38)(16 24 20 22)(17 21 19 25)(18 23)(26 34 30 32)(27 31 29 35)(28 33)

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

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

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

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

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

35 conjugacy classes

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	4E	4F	4G	4H	5	6A	6B	6C	6D	6E	6F	10	12A	···	12H	15A	15B	20	30A	30B
order	1	2	2	2	3	3	4	4	4	4	4	4	4	4	5	6	6	6	6	6	6	10	12	···	12	15	15	20	30	30
size	1	1	5	5	4	4	5	5	5	5	6	30	30	30	4	4	4	20	20	20	20	4	20	···	20	16	16	24	16	16

35 irreducible representations

dim	1	1	1	1	1	1	12	2	2	2	3	3	3	4	4	8	8
type	+	+					+	-			+	+		+		-
image	C₁	C₂	C₃	C₄	C₆	C₁₂	A₄×F₅	SL₂(𝔽₃)	SL₂(𝔽₃)	C₄.A₄	A₄	C₂×A₄	C₄×A₄	F₅	C₃×F₅	F₅×SL₂(𝔽₃)	F₅×SL₂(𝔽₃)
kernel	F₅×SL₂(𝔽₃)	D₅×SL₂(𝔽₃)	Q₈×F₅	C₅×SL₂(𝔽₃)	Q₈×D₅	C₅×Q₈	C₂	F₅	F₅	D₅	C₂×F₅	D₁₀	C₁₀	SL₂(𝔽₃)	Q₈	C₁	C₁
# reps	1	1	2	2	2	4	1	2	4	6	1	1	2	1	2	1	2

Matrix representation of F₅×SL₂(𝔽₃) ►in GL₆(𝔽₆₁)

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

11	0	0	0	0	0
0	11	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

0	1	0	0	0	0
60	0	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

48	14	0	0	0	0
14	13	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

1	14	0	0	0	0
0	13	0	0	0	0
0	0	47	0	0	0
0	0	0	47	0	0
0	0	0	0	47	0
0	0	0	0	0	47

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[48,14,0,0,0,0,14,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,14,13,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,47] >;

F₅×SL₂(𝔽₃) in GAP, Magma, Sage, TeX

F_5\times {\rm SL}_2({\mathbb F}_3)

% in TeX

G:=Group("F5xSL(2,3)");

// GroupNames label

G:=SmallGroup(480,965);

// by ID

G=gap.SmallGroup(480,965);

# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-5,-2,42,514,584,221,795,382,4037,1363]);

// Polycyclic

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

// generators/relations

G = F5×SL2(𝔽3) order 480 = 25·3·5

Direct product of F5 and SL2(𝔽3)

G = F₅×SL₂(𝔽₃) order 480 = 2⁵·3·5

Direct product of F₅ and SL₂(𝔽₃)