#27551071

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2634 ⋊ C₂
Sylow	C₅₃ ⋊ #34604

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2634	C₂
prime radical	C₂³×D₅₃	C₂⁴
hypercenter	#2634	D₅₃
hypocenter	C₅₃	#34604
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2634

1536 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:

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=h,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch,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, 336547);