C₇×#393⋊C₃ #13055049

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₇×#393⋊C₃
unambiguous semidirect product	#393⋊(C₃×C₇)

	subgroup	G/H
center, hypercenter	C₂×C₇	#120⋊C₃
commutator, cosocle, hypocenter	C₂²×Q₈	C₂×C₃×C₇
Frattini	C₂³	C₂×C₇×A₄
Fitting	C₇×#393	C₃
socle	C₂³×C₇	C₂×A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂×C₇

114 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4,5,6,7)(8,9)(10,13,14,12,11,15)(17,18,20)(21,23,22);(9,15)(12,13)(16,20,19,22)(17,23,21,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4338);