C₂×C₃×C₄⋊Q₈ #5861

order	192
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×C₃×C₄⋊Q₈

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

290 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1420);