C₃×#3253 #85117

order	384
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

430 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17693);