#27276405

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8485

Subgroups

	subgroup	G/H
center	C₂²	#16810479
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3563	C₂
socle	C₂²×C₅₃	#379
prime radical	#121×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#3563	D₅₃
hypocenter	C₅₃	#8485
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3563

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61881);