#27551464

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2657 ⋊ C₂
Sylow	C₅₃ ⋊ #34641

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2657	C₂
hypercenter	#2657	D₅₃
hypocenter	C₅₃	#34641
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2657

3694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 77 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)(62,65,66,72)(63,69,68,75)(64,71,70,76)(67,74,73,77);(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)(63,68)(67,73)(69,75)(74,77);(58,59,60,61)(64,70)(67,73)(71,76)(74,77);(54,55,56,57)(62,64)(63,67)(65,71)(66,70)(68,73)(69,74)(72,76)(75,77);(55,57)(59,61)(62,63)(64,67)(65,69)(66,68)(70,73)(71,74)(72,75)(76,77)

Polycyclic presentation:

2^8,53|b^2=f,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dh,c^b=ch,i^d=i^52,i^e=i^52

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 336940);