#27590690

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2844 ⋊ C₂
Sylow	C₅₃ ⋊ #36691

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2844	C₂
prime radical	C₂³×D₅₃	C₂⁴
hypercenter	#2844	D₅₃
hypocenter	C₅₃	#36691
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2844

1474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 85 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)(70,73,74,81)(71,77,76,83)(72,80,79,85)(75,82,78,84);(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)(62,64)(63,67)(65,68)(66,69)(71,76)(75,78)(77,83)(82,84);(54,56,57,61)(55,58,59,60)(63,66)(64,68)(72,79)(75,78)(80,85)(82,84);(62,63,65,66)(64,67,68,69)(70,72,74,79)(71,75,76,78)(73,80,81,85)(77,82,83,84);(54,55,57,59)(56,60,61,58)(64,68)(67,69)(70,71)(72,78)(73,77)(74,76)(75,79)(80,84)(81,83)(82,85)

Polycyclic presentation:

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

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 376166);