#71384

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2252

Subgroups

	subgroup	G/H
center	C₂²	C₂×#213
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×#500	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	#500	D₃
hypocenter	C₃	#2252
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

128 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)(12,13,18,24,15,19,22,25,16,20,26,27,21,23,14,17);(1,2,3)(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)(13,17)(14,18)(15,21)(19,27)(20,25)(22,26)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10709133, #10709134, #10709135, …
maximal normal subgroup of	#10709133, #10709134, #10709135, …
maximal quotient of	#10711620, #10713214, #10713787, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3960);