#66436

order	324
isoclinism	isoclinic class of C₉⋊A₄
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×C₉ ⋊ C₃
Sylow	C₂² ⋊ #737

Subgroups

	subgroup	G/H
center	C₉	C₃×A₄
commutator, socle, cosocle	C₂²×C₃	C₃³
Frattini	C₃	C₃²×A₄
Fitting	C₂²×C₃×C₉	C₃
prime radical	C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×C₉	A₄
hypocenter	C₂²	#737
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₉ ◁ C₃×C₉

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 31 points:

(1,4)(2,3)(6,11,12)(9,17,21)(10,19,20)(16,25,28)(18,26,27)(24,30,31);(5,7,13,8,14,22,15,23,29)(6,10,18,12,20,27,11,19,26)(9,16,24,17,25,30,21,28,31);(2,3,4)(5,6,9)(7,10,16)(8,12,17)(11,21,15)(13,18,24)(14,20,25)(19,28,23)(22,27,30)(26,31,29)

Polycyclic presentation:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10620044⋊C₂, #72276182
maximal subgroup of	#10620044, #10620045, #10620070, …
maximal normal subgroup of	#10620044, #10620045, #10620070, …
maximal quotient of	#10619912, C₂×#66436, #11757759, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 128);