C₃×#2198 #72913

order	384
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂³×C₃
prime radical	C₃×#453	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃ ◁ C₂×C₃×D₄ ◁ G

246 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5489);