#27590870

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2689 ⋊ C₂
Sylow	C₅₃ ⋊ #36705

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2689	C₂
prime radical	C₂×#16811212	C₂
hypercenter	#2689	D₅₃
hypocenter	C₅₃	#36705
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2689

2406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 5 generators on 81 points:

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

Polycyclic presentation:

2^8,53|b^2=f,c^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,i^a=i^52,c^b=ch,d^b=dfh,i^e=i^52

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 376346);