#13062002

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

Aliases

No aliases for this group.

Splittings

derived

SL(2,𝔽₇) ⋊ C₂²

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂	C₂×SO(3,𝔽₇)
commutator, cosocle, hypocenter	SL(2,𝔽₇)	C₂²
Fitting, hypercenter	C₄	SO(3,𝔽₇)
LCS	G ▷ SL(2,𝔽₇)
UCS	C₁ ◁ C₂ ◁ C₄

93 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:

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

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, 11291);