#27551044

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #34602

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2618	C₂
prime radical	C₂×#121×C₅₃	C₂²
hypercenter	#2618	D₅₃
hypocenter	C₅₃	#34602
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2618

2294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

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)(58,60,61,64)(59,63,62,65);(67,73)(71,79)(72,81)(78,80);(55,57)(66,69,70,77)(67,72,73,81)(68,74,75,76)(71,78,79,80);(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)(59,62)(63,65)(66,68,70,75)(67,71,73,79)(69,76,77,74)(72,80,81,78);(54,55)(56,57)(58,59)(60,63)(61,62)(64,65)(66,67)(68,71)(69,72)(70,73)(74,78)(75,79)(76,80)(77,81)

Polycyclic presentation:

2^8,53|b^2=h,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch,i^b=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, 336520);