C₃×#2683 #84547

order	384
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

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

592 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 224 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17123);