#13062009

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

non-solvable
monolithic
stem

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂	C₂×SO(3,𝔽₇)
commutator, cosocle, hypocenter	SL(2,𝔽₇)	C₂²
Fitting, hypercenter	C₂²	SO(3,𝔽₇)
LCS	G ▷ SL(2,𝔽₇)
UCS	C₁ ◁ C₂ ◁ C₂²

83 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

23 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11298);