C₇×#399⋊C₃ #13055052

order	1344
isoclinism	isoclinic class of #392⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×#399⋊C₃
unambiguous semidirect product	#399⋊(C₃×C₇)

	subgroup	G/H
center, hypercenter	C₂×C₇	#127⋊C₃
commutator, cosocle, hypocenter	#95	C₂×C₃×C₇
Frattini	C₂³	C₂×C₇×A₄
Fitting	C₇×#399	C₃
socle	C₂³×C₇	C₂×A₄
LCS	G ▷ #95
UCS	C₁ ◁ C₂×C₇

102 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4341);