#71360

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1707

Subgroups

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#500	C₂
socle, prime radical	C₂²×C₃	#127
hypercenter	#500	D₃
hypocenter	C₃	#1707
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

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

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

Polycyclic presentation:

2^7,3|a^2=f,b^2=df,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,h^c=h^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10709333, #10709334, #10709340, …
maximal normal subgroup of	#10709333, #10709334, #10709340, …
maximal quotient of	#10688955, #10694922, #10694933, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3936);