#66357

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×C₉ ⋊ C₃
Sylow	C₂² ⋊ #731

Subgroups

	subgroup	G/H
center	C₃	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃²
Frattini	C₃²	C₃×A₄
Fitting	C₂²×C₃×C₉	C₃
socle	C₂²×C₃	H(3,𝔽₃)
prime radical	C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×C₉	A₄
hypocenter	C₂²	#731
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉

38 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)(5,7,14,8,15,25,16,26,31)(6,10,20,12,21,24,11,13,23)(9,17,27,18,19,28,22,29,30);(2,3,4)(5,6,9)(7,13,17)(8,12,18)(10,19,15)(11,22,16)(14,24,27)(20,30,31)(21,29,26)(23,28,25)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10619751, #13047010, #10619741⋊C₂, …
maximal subgroup of	#10619741, #10619751, #10619769, …
maximal normal subgroup of	#10619741, #10619751, #10619769, …
maximal quotient of	#10619574, C₂×#66357, #11757399, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 49);