C₇×#126⋊C₃ #10621167

order	672
isoclinism	isoclinic class of #126⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×#126⋊C₃
unambiguous semidirect product	#126⋊(C₃×C₇)

	subgroup	G/H
center, hypercenter	C₇	#126⋊C₃
commutator, cosocle, hypocenter	C₄²	C₂×C₃×C₇
Frattini	C₂²	C₂×C₇×A₄
Fitting	C₇×#126	C₃
socle	C₂²×C₇	C₂×A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₇

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 749);