#66358

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×C₉ ⋊ 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×C₃×C₉	C₃
socle	C₂²×C₃	H(3,𝔽₃)
hypercenter	C₃×C₉	A₄
hypocenter	C₂²	#732
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉

42 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)(5,7,13,8,14,25,15,26,31)(6,10,20,11,12,22,21,23,24)(9,16,27,18,28,30,17,19,29);(2,3,4)(5,6,9)(7,12,16)(8,11,18)(10,19,26)(13,24,27)(14,23,28)(15,21,17)(20,30,25)(22,29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10619750, #13047011, #10619740⋊C₂, …
maximal subgroup of	#10619740, #10619750, #10619768, …
maximal normal subgroup of	#10619740, #10619750, #10619768, …
maximal quotient of	#10619575, C₂×#66358, #11757396, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 50);