#92392

order	486
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

H(5,𝔽₃) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, socle	C₃	C₃²×C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂×C₃²
Fitting	H(5,𝔽₃)	C₂
hypercenter	H(3,𝔽₃)	C₃⋊D₃
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃ ◁ H(3,𝔽₃)

238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#14519129, #14519160, #72228599, …
maximal subgroup of	#1088⋊C₃², #11758049, #66432⋊C₃, …
maximal normal subgroup of	#1088⋊C₃², #11758049, #66432⋊C₃, …
maximal quotient of	#11758013, C₂×#92392, #13091378, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 237);