#66362

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×H(3,𝔽₃) ⋊ C₃
Sylow	C₂² ⋊ #732

Subgroups

	subgroup	G/H
center	C₃	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃²
Frattini	C₃²	C₃×A₄
Fitting	C₂²×H(3,𝔽₃)	C₃
socle	C₂²×C₃	H(3,𝔽₃)
hypercenter	H(3,𝔽₃)	A₄
hypocenter	C₂²	#732
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ H(3,𝔽₃)

43 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)(6,10,20)(8,16,25)(9,18,27)(11,22,30)(12,13,24)(15,26,17)(19,23,31)(21,29,28);(2,3,4)(5,6,9)(7,13,23)(8,12,19)(10,18,15)(11,21,17)(14,24,27)(16,22,29)(20,28,26)(25,30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10619755, #13047015, #10619743⋊C₂, …
maximal subgroup of	#10619743, #10619755, #10619771, …
maximal normal subgroup of	#10619743, #10619755, #10619771, …
maximal quotient of	#10619579, C₂×#66362, #11757401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 54);