#27276269

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1516 ⋊ C₂
Sylow	C₅₃ ⋊ #8463

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1516	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288432	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1516	D₅₃
hypocenter	C₅₃	#8463
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1516

718 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:

(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)(54,57,58,65)(55,60,61,67)(56,63,64,69)(59,66,62,68)(70,73)(71,78)(72,83)(74,89)(75,87)(76,88)(77,90)(79,95)(80,93)(81,94)(82,96)(84,100)(85,99)(86,98)(91,101)(92,97);(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)(55,59)(57,63)(58,64)(60,66)(61,62)(65,69)(67,68)(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);(54,55)(56,62,64,59)(57,60)(58,61)(63,68,69,66)(65,67)(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=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,i^c=i^52

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, 61745);