C₂×#983 #5917

order	192
automorphism	Aut(G) = S₄×#120 (order 768) · Inn(G) = C₂×S₄ · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×Q₈ ⋊ D₃

	subgroup	G/H
center, socle	C₂²	C₂×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂³
Frattini	C₂	C₂²×S₄
Fitting	C₂²×Q₈	D₃
hypercenter	C₂³	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂³

153 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1476);