#27276512

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8502

Subgroups

	subgroup	G/H
center	C₂²	#16810479
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3547	C₂
socle	C₂²×C₅₃	#379
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3547	D₅₃
hypocenter	C₅₃	#8502
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3547

672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 101 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)(54,56)(55,60)(57,66)(58,65)(59,62)(61,68)(63,69)(64,67)(70,73,76,88)(71,78,81,94)(72,83,85,99)(74,89,75,87)(77,90,91,101)(79,95,80,93)(82,96,92,97)(84,100,86,98);(54,56)(55,59)(57,64)(58,65)(60,62)(61,63)(66,67)(68,69)(70,72)(71,77)(73,83)(74,86)(75,84)(76,85)(78,90)(79,92)(80,82)(81,91)(87,100)(88,99)(89,98)(93,96)(94,101)(95,97);(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)(56,63,65,59)(57,62)(58,61)(60,64,68,66)(67,69)(70,71,75,80,76,81,74,79)(72,82,86,77,85,92,84,91)(73,78,87,93,88,94,89,95)(83,97,98,101,99,96,100,90)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,d^2=h,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,i^a=i^52,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg

Conjugacy classes & character table

1022 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61988);