#71298

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1713

Subgroups

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#494	C₂
socle, prime radical	C₂²×C₃	#127
hypercenter	#494	D₃
hypocenter	C₃	#1713
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

192 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:

(2,3)(4,5,9,11,7,10,6,8);(1,2,3)(12,14,21,25,16,23,15,22)(13,17,24,27,19,26,18,20);(5,8)(6,9)(10,11)(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=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^c=h^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10683050, #10708594, #10708595, …
maximal normal subgroup of	#10683050, #10708594, #10708595, …
maximal quotient of	#10688967, #10694943, #10694953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3874);