#66364

order	324
automorphism	Aut(G) = C₃×A₄⋊#272 (order 1944) · Inn(G) = C₂²⋊H(3,𝔽₃) · Out(G) = C₃×D₃
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Sylow

C₂² ⋊ #733

Subgroups

	subgroup	G/H
center	C₃	C₂²⋊H(3,𝔽₃)
commutator, prime radical, cosocle	C₂²×C₃²	C₃²
Frattini	C₃²	C₃×A₄
Fitting	C₂²×3₋¹⁺²	C₃
socle	C₂²×C₃	H(3,𝔽₃)
hypercenter	3₋¹⁺²	A₄
hypocenter	C₂²	#733
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ 3₋¹⁺²

34 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)(5,7,13,8,15,25,16,14,26)(6,10,20,11,21,30,22,12,24)(9,17,27,18,23,31,28,19,29);(2,3,4)(5,6,9,8,11,18,16,22,28)(7,12,23,15,10,19,14,21,17)(13,20,27,25,30,31,26,24,29)

Polycyclic presentation:

3^4,2^2|a^3=d,b^3=d,b^a=bc,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	#66364⋊C₂, #66364⋊C₄, C₂×#66364⋊C₂, …
maximal subgroup of	#66364⋊C₂, C₂×#66364, #11757798, …
maximal normal subgroup of	#66364⋊C₂, C₂×#66364, #11757798, …
maximal quotient of	#10619581, C₂×#66364, #11757399, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 56);