#66361

order	324
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₂² ⋊ #731

Subgroups

	subgroup	G/H
center	C₃	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃²
Frattini	C₃²	C₃×A₄
Fitting, prime radical	C₂²×H(3,𝔽₃)	C₃
socle	C₂²×C₃	H(3,𝔽₃)
hypercenter	H(3,𝔽₃)	A₄
hypocenter	C₂²	#731
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ H(3,𝔽₃)

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 31 points:

(1,4)(2,3)(5,7,13)(6,10,19)(8,14,25)(9,16,27)(11,12,22)(15,26,24)(17,18,29)(20,23,30)(21,31,28);(2,3,4)(5,6,9,8,11,17,15,20,28)(7,12,21,14,23,16,26,10,18)(13,19,27,25,22,29,24,30,31)

Polycyclic presentation:

3^4,2^2|a^3=d,b^a=bc,e^a=f,f^a=ef,c^b=cd^2

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10619756, #13047016, #10619744⋊C₂, …
maximal subgroup of	#10619744, #10619756, #10619772, …
maximal normal subgroup of	#10619744, #10619756, #10619772, …
maximal quotient of	#10619578, C₂×#66361, #11757399, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 53);