C₃×D₄×Q₈ #5879

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 6144
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

C₃×D₄×Q₈

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂×C₃×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

280 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1438);