C₇×#5934 #13062131

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₂³

Aliases

Fitting	C₇×H(5,𝔽₂) ⋊ D₃
derived	#997 ⋊ C₂×C₇

	subgroup	G/H
center, socle, hypercenter	C₂×C₇	#1020
commutator, cosocle, hypocenter	#997	C₂×C₇
Frattini	C₂	C₇×#1020
Fitting	C₇×H(5,𝔽₂)	D₃
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₇

196 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,3,4,5,6,7)(8,9,12,14)(10,11,13,15);(9,10,12)(13,14,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11420);