#92319

order	486
isoclinism	isoclinic class of #272
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#6903 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₂₇	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₂×C₉
Frattini	C₉	C₃×C₃⋊D₃
Fitting	#6903	C₂
socle	C₃	C₉×C₃⋊D₃
prime radical	#272	C₉
cosocle	#737	C₂×C₃
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂₇

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 3 generators on 81 points:

(1,7,35,12,43,41,53,2,17,5,22,57,32,67,6,10,9,37,16,47,75,3,20,19,28,26,61)(4,44,56,54,66,63,51,23,36,33,46,80,30,79,18,50,25,76,14,69,73,29,42,62,52,49,59,13,21,72,31,78,39,71,8,58,15,68,74,55,65,38,27,48,60,34,45,81,11,64,40,70,24,77);(4,14,15)(8,24,25)(11,30,31)(13,33,34)(18,39,40)(21,45,46)(23,48,49)(27,51,52)(29,54,55)(36,59,60)(38,62,63)(42,65,66)(44,68,69)(50,70,71)(56,73,74)(58,76,77)(64,78,79)(72,80,81);(1,4,13)(2,8,23)(3,11,29)(5,15,33)(6,18,38)(7,21,44)(9,25,48)(10,27,50)(12,31,54)(14,34,16)(17,36,58)(19,40,62)(20,42,64)(22,46,68)(24,49,26)(28,52,70)(30,55,32)(35,56,72)(37,60,76)(39,63,41)(43,66,78)(45,69,47)(51,71,53)(57,74,80)(59,77,61)(65,79,67)(73,81,75)

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11757686⋊C₂
maximal subgroup of	#11757686, #92318⋊C₂, C₂×#92319, …
maximal normal subgroup of	#11757686, #92318⋊C₂, C₂×#92319, …
maximal quotient of	#11757623, C₂×#92319, #13090866, …
automorphism group of	C₈₁⋊C₃, C₂×C₈₁⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 164);