C₂×C₃×#480 #84949

order	384
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ G

512 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(3,4,5);(6,7)(8,12)(9,10)(11,13)(15,18)(16,20)(19,21);(1,2)(6,8)(7,11)(9,12)(10,13);(6,9,10,7)(8,12,13,11)(14,15)(16,21)(17,19)(18,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17525);