C₃×#2032 #72747

order	384
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#393
commutator, Frattini, cosocle	C₂⁴	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂⁴×C₃ ◁ G

278 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18,19);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5323);