C₂×#994 #5943

order	192
automorphism	Aut(G) = S₄×#1020⋊C₂ (order 4608) · Inn(G) = C₂²×A₄ · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂²	C₂²×A₄
commutator, cosocle, hypocenter	Q₈	C₂³×C₃
Frattini	C₂	C₂³×A₄
Fitting	C₂×H(5,𝔽₂)	C₃
hypercenter	C₂×Q₈	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

208 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 26 points:

(1,2)(3,13,23,11,6,21,8,20,14,5,17,26)(4,18,22,7,10,25,12,16,19,9,15,24);(5,11)(6,16)(7,9)(10,20)(13,15)(14,26)(17,18)(19,25)(21,23)(22,24);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1502);