#27591317

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #36732

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₂×C₅₃×#381	C₂
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂×#381	D₅₃
hypocenter	C₅₃	#36732
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#381

1770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 5 generators on 85 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(63,67)(66,69)(71,78)(76,79)(77,82)(84,85);(64,68)(66,69)(70,74,75,73)(71,77,78,82)(72,80,81,83)(76,85,79,84);(55,59)(58,61)(64,68)(66,69)(70,73,75,74)(71,77,78,82)(72,80,81,83)(76,84,79,85);(54,56,57,60)(55,58,59,61)(62,64)(63,66)(65,68)(67,69)(70,72,75,81)(71,76,78,79)(73,80,74,83)(77,84,82,85);(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)(56,58)(57,59)(60,61)(62,63)(64,66)(65,67)(68,69)(70,71,75,78)(72,79,81,76)(73,77,74,82)(80,85,83,84)

Polycyclic presentation:

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

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 376793);