#27592054

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2789 ⋊ C₂
Sylow	C₅₃ ⋊ #36762

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2789	C₂
prime radical	C₂³×D₅₃	C₂⁴
hypercenter	#2789	D₅₃
hypocenter	C₅₃	#36762
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2789

1482 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)(55,59)(58,61)(71,77)(75,78)(76,81)(84,85);(56,60)(58,61)(70,73,74,83)(71,76,77,81)(72,79,80,82)(75,84,78,85);(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)(56,60)(58,61)(62,65,66,64)(63,67,68,69);(54,56)(55,58)(57,60)(59,61)(62,64,66,65)(63,67,68,69)(70,72,74,80)(71,75,77,78)(73,82,83,79)(76,85,81,84);(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)(54,55)(56,58)(57,59)(60,61)(62,63)(64,67)(65,69)(66,68)(70,71,74,77)(72,78,80,75)(73,81,83,76)(79,84,82,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=fg,c^2=g,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,i^c=i^52,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, 377530);