#66365

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

Fitting	C₂²×3₋¹⁺² ⋊ C₃
Sylow	C₂² ⋊ #731

Subgroups

	subgroup	G/H
center	C₃	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃²
Frattini	C₃²	C₃×A₄
Fitting	C₂²×3₋¹⁺²	C₃
socle	C₂²×C₃	H(3,𝔽₃)
prime radical	C₂²⋊H(3,𝔽₃)	C₃
hypercenter	3₋¹⁺²	A₄
hypocenter	C₂²	#731
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ 3₋¹⁺²

36 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)(5,7,12,8,14,24,15,13,26)(6,10,20,11,21,30,22,25,23)(9,16,27,18,19,29,17,28,31);(2,3,4)(5,6,9)(7,10,19)(8,11,18)(12,23,27)(13,25,16)(14,21,28)(15,22,17)(20,29,24)(26,30,31)

Polycyclic presentation:

3^4,2^2|b^3=d,b^a=bc,c^a=cd^2,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	#66365⋊C₂, #66365⋊C₄, C₂×#66365⋊C₂, …
maximal subgroup of	#66365⋊C₂, C₂×#66365, #11757798, …
maximal normal subgroup of	#66365⋊C₂, C₂×#66365, #11757798, …
maximal quotient of	#10619582, C₂×#66365, #11757400, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 57);