C₇×#127⋊C₃ #10621168

order	672
isoclinism	isoclinic class of #127⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₄

Aliases

direct product	C₇×#127⋊C₃
unambiguous semidirect product	#127⋊(C₃×C₇)

	subgroup	G/H
center, hypercenter	C₇	#127⋊C₃
commutator, cosocle, hypocenter	C₄²	C₂×C₃×C₇
Frattini	C₂²	C₂×C₇×A₄
Fitting	C₇×#127	C₃
socle	C₂²×C₇	C₂×A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₇

56 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 84

A minimal faithful permutation representation — 2 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 750);