#27475965

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#471 ⋊ C₂
Sylow	C₅₃ ⋊ #28499

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#471	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20285177	C₂
hypercenter	C₂×#471	D₅₃
hypocenter	C₅₃	#28499
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#471

1004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(58,62,63,74,65,73,64,60)(59,68,69,66,70,71,79,82)(61,76,77,87,78,88,80,72)(67,83,84,81,85,86,75,89);(55,57)(58,61)(59,67)(60,72)(62,76)(63,77)(64,80)(65,78)(66,81)(68,83)(69,84)(70,85)(71,86)(73,88)(74,87)(75,79)(82,89);(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)(54,55,56,57)(58,60,63,62,65,74,64,73)(59,66,69,71,70,82,79,68)(61,72,77,76,78,87,80,88)(67,81,84,86,85,89,75,83);(58,59,64,69,65,70,63,79)(60,71,73,82,74,68,62,66)(61,75,77,85,78,84,80,67)(72,81,76,83,87,89,88,86)

Polycyclic presentation:

2^8,53|a^2=f,b^2=eg,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,i^b=i^52,f^c=fh

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 261441);