C₂×#857 #5385

order	192
isoclinism	isoclinic class of #857
automorphism	Aut(G) = C₂×#857⋊C₂ (order 384) · Inn(G) = #857 · Out(G) = C₂²
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

Fitting	C₂×C₄² ⋊ D₃
derived	C₄²⋊C₃ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	#857
commutator, cosocle, hypocenter	C₄²⋊C₃	C₂²
Frattini	C₂²	C₂×S₄
Fitting	C₂×C₄²	D₃
socle	C₂³	S₄
LCS	G ▷ C₄²⋊C₃
UCS	C₁ ◁ C₂

84 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 12 points:

(3,5)(4,9,10,8,6,12,7,11);(1,6,9,2,4,12)(3,10,8,5,7,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 944);