#6729

order	240
isoclinism	isoclinic class of #6728
automorphism	Aut(G) = C₂×S₅ (order 240) · Inn(G) = S₅ · Out(G) = C₂
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived

SL(2,𝔽₅) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, Fitting, socle, hypercenter	C₂	S₅
commutator, cosocle, hypocenter	SL(2,𝔽₅)	C₂
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂

25 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 40 points:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	1	20	20	20	20	20	24	24	30	30	30
order	1	2	2	3	6	6	6	5	10	4	8	8
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	-1	1	1	-1	-1	1	1	1	-1	-1
χ₃	4	4	2	1	1	-1	-1	-1	-1	0	0	0
χ₄	4	4	-2	1	1	1	1	-1	-1	0	0	0
χ₅	4	-4	0	1	-1	$- i \sqrt{3}$	$i \sqrt{3}$	-1	1	0	0	0
χ₆	4	-4	0	1	-1	$i \sqrt{3}$	$- i \sqrt{3}$	-1	1	0	0	0
χ₇	4	-4	0	-2	2	0	0	-1	1	0	0	0
χ₈	5	5	1	-1	-1	1	1	0	0	1	-1	-1
χ₉	5	5	-1	-1	-1	-1	-1	0	0	1	1	1
χ₁₀	6	6	0	0	0	0	0	1	1	-2	0	0
χ₁₁	6	-6	0	0	0	0	0	1	-1	0	$- i \sqrt{2}$	$i \sqrt{2}$
χ₁₂	6	-6	0	0	0	0	0	1	-1	0	$i \sqrt{2}$	$- i \sqrt{2}$

Here i is the imaginary unit (i² = −1).

Relationships

maximal subgroup of	#91870, #91872, C₂×#6729, …
maximal normal subgroup of	#91870, #91872, C₂×#6729, …
maximal quotient of	#91143, C₂×#6729, C₃×#6729, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 90);