Q₈×Dic₆ #5566

order	192
automorphism	Aut(G) order 18432
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

C₃ ⋊ Q₈²

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×C₄×Q₈	C₂
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	Q₈²
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

212 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1125);