#92404

order	486
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃⁵

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(5,𝔽₃) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₃	C₃⋊(C₃⋊(C₃⋊D₃))
commutator, Fitting, cosocle, hypocenter	H(5,𝔽₃)	C₂
LCS	G ▷ H(5,𝔽₃)
UCS	C₁ ◁ C₃

586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 5 generators on 27 points:

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

Polycyclic presentation:

2,3^5|b^a=b^2,c^a=c^2,d^a=d^2,e^a=e^2,e^b=ef,d^c=df

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#14519140, #14519141, #14519142, …
maximal subgroup of	#11758051, #11758052, #11758086, …
maximal normal subgroup of	#11758051, #11758052, #11758086, …
maximal quotient of	#11758025, C₂×#92404, #13091382, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 249);