#92204

order	486
isoclinism	isoclinic class of #92204
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#6862 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	#6862	C₂
Frattini	C₃³	C₃⋊D₃
socle	C₃²	#272
LCS	G ▷ #6862
UCS	C₁

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

Polycyclic presentation:

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

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11757372⋊C₂, #14516414
maximal subgroup of	#11757372, #11757373, #92167⋊C₂, …
maximal normal subgroup of	#11757372, #11757373, #92167⋊C₂, …
maximal quotient of	#11757327, C₂×#92204, #13090453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 49);