C₃×#1649 #72364

order	384
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#1649

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃
prime radical	C₂×C₃×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

234 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,19,12,21)(13,23,24,17)(15,22,16,20)(25,26,27);(2,5)(3,8)(6,7)(9,11,14,12)(10,15,18,16)(13,22,24,20)(17,21,23,19);(9,12,14,11)(10,16,18,15)(13,20,24,22)(17,19,23,21)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#1649, #10771035, #10771144, …
maximal subgroup of	C₂×C₃×#1649, C₃×#18503, C₃×#18516, …
maximal normal subgroup of	C₂×C₃×#1649, C₃×#18503, C₃×#18516, …
maximal quotient of	C₃×#7756, C₃×#7757, C₃×#7763, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4940);