#27366258

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3234 ⋊ C₂
Sylow	C₅₃ ⋊ #19092

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3234	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20285752	C₂
hypercenter	#3234	D₅₃
hypocenter	C₅₃	#19092
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3234

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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)(55,62)(60,65)(61,68)(67,69)(71,78)(72,82)(77,92)(79,86)(81,98)(83,87)(93,95)(99,100);(54,56,59,64)(55,61,62,68)(57,63,58,66)(60,67,65,69)(70,73,75,85)(71,77,78,92)(72,81,82,98)(74,88,89,84)(76,90,91,101)(79,95,86,93)(80,96,94,97)(83,100,87,99);(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)(56,61)(57,65)(58,60)(59,62)(63,69)(64,68)(66,67)(70,72,75,82)(71,76,78,91)(73,81,85,98)(74,87,89,83)(77,90,92,101)(79,94,86,80)(84,100,88,99)(93,96,95,97);(55,60)(56,63)(57,58)(62,65)(64,66)(67,69)(70,71)(72,80)(73,77)(74,86)(75,78)(76,83)(79,89)(81,97)(82,94)(84,95)(85,92)(87,91)(88,93)(90,99)(96,98)(100,101)

Polycyclic presentation:

2^8,53|b^2=h,c^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,i^b=i^52,e^c=eh,f^c=fgh

Conjugacy classes & character table

707 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 151734);