#27547393

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2370 ⋊ C₂
Sylow	C₅₃ ⋊ #34403

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#2370	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂³×D₅₃	C₂⁴
hypercenter	#2370	D₅₃
hypocenter	C₅₃	#34403
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2370

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 89 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)(58,62,63,61)(59,66,65,70)(60,69,67,68)(64,71,72,73)(74,78,79,77)(75,81,82,87)(76,86,85,84)(80,88,83,89);(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)(58,61,63,62)(59,66,65,70)(60,68,67,69)(64,71,72,73)(76,85)(80,83)(84,86)(88,89);(59,65)(60,67)(66,70)(68,69)(74,77,79,78)(75,81,82,87)(76,84,85,86)(80,88,83,89);(58,60)(59,64)(61,68)(62,69)(63,67)(65,72)(66,71)(70,73)(74,76,79,85)(75,80,82,83)(77,84,78,86)(81,88,87,89);(54,55,56,57)(58,59)(60,64)(61,70)(62,66)(63,65)(67,72)(68,73)(69,71)(74,75)(76,83)(77,81)(78,87)(79,82)(80,85)(84,89)(86,88)

Polycyclic presentation:

2^8,53|a^2=h,b^2=f,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,i^e=i^52

Conjugacy classes & character table

1376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 332869);