C₃×#1750 #72465

order	384
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃×#1750

Subgroups

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

336 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,4,3)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,27);(6,8)(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(1,3,4,2)(10,15)(14,20)(16,22)(21,24)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10682215, #10682221, #10686707, …
maximal subgroup of	C₃×#7214, C₃×#7216, C₃×#8348, …
maximal normal subgroup of	C₃×#7214, C₃×#7216, C₃×#8348, …
maximal quotient of	C₃×#8148, C₃×#8149, C₃×#8154, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5041);