#66410

order	324
isoclinism	isoclinic class of #272
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

#737 ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₉	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₃×C₄
Frattini, socle	C₂×C₃	C₃×C₃⋊D₃
Fitting	C₂×#737	C₂
prime radical, cosocle	C₂×H(3,𝔽₃)	C₂×C₃
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₉

67 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 108

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13048270, #13048271, #13048272
maximal subgroup of	#10619833, #10619905, #4207⋊C₄, …
maximal normal subgroup of	#10619833, #10619905, #4207⋊C₄, …
maximal quotient of	#10619725, C₂×#66410, #11757599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 102);