C₇×#857 #10620696

order	672
isoclinism	isoclinic class of #857
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₂

Aliases

Fitting	C₄²×C₇ ⋊ D₃
derived	C₄²⋊C₃ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₇	#857
commutator, cosocle, hypocenter	C₄²⋊C₃	C₂×C₇
Frattini	C₂²	C₇×S₄
Fitting	C₄²×C₇	D₃
socle	C₂²×C₇	S₄
LCS	G ▷ C₄²⋊C₃
UCS	C₁ ◁ C₇

48 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)(8,10,11,15)(9,19,18,17,13,16,14,12);(8,9,12)(10,14,16)(11,13,17)(15,18,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 278);