#71334

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#498 ⋊ C₂
Sylow	C₃ ⋊ #1708

Subgroups

	subgroup	G/H
center	C₂²	#940
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#498	C₂
socle	C₂²×C₃	#127
prime radical	#874	C₂²
hypercenter	#498	D₃
hypocenter	C₃	#1708
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

190 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^7,3|b^2=d,c^2=ef,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	#10708950, D₃×#1708, C₂×#71334, …
maximal normal subgroup of	#10708950, D₃×#1708, C₂×#71334, …
maximal quotient of	#10688959, #10694923, #10694925, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3910);