#92312

order	486
isoclinism	isoclinic class of #92312
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₉

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#6898 ⋊ C₂
derived	C₉² ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉²	C₂×C₃
Frattini, socle	C₃²	C₃×C₃⋊D₃
Fitting	#6898	C₂
LCS	G ▷ C₉²
UCS	C₁

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 3 generators on 54 points:

(2,5)(3,19,21,9,10,20)(4,11)(6,8)(7,24,26,15,16,25)(12,27,14,17,22,23)(13,18)(29,40,35,32,33,42)(30,36)(31,39)(34,53,46,44,43,51)(37,38)(41,45,52,47,50,54)(48,49);(1,3,9)(2,7,17)(4,10,20)(5,12,15)(6,14,24)(8,16,23)(11,21,19)(13,22,25)(18,26,27)(28,49,36,39,38,37,31,30,48)(29,43,47,33,46,45,35,34,54)(32,52,44,42,50,51,40,41,53);(1,18,5,11,8,6,4,2,13)(3,26,12,21,16,14,10,7,22)(9,27,15,19,23,24,20,17,25)(28,29,32)(30,34,41)(31,35,40)(33,42,39)(36,47,44)(37,45,51)(38,46,50)(43,52,49)(48,54,53)

Polycyclic presentation:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11757683⋊C₂, #14517943
maximal subgroup of	#11757683, #11757691, #92311⋊C₂, …
maximal normal subgroup of	#11757683, #11757691, #92311⋊C₂, …
maximal quotient of	#11757616, C₂×#92312, #13090632, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 157);