#27281667

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1517 ⋊ C₂
Sylow	C₅₃ ⋊ #9432

Subgroups

	subgroup	G/H
center	C₂²	#16810499
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1517	C₂
socle	C₂²×C₅₃	#386
prime radical	#20285407	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1517	D₅₃
hypocenter	C₅₃	#9432
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1517

436 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 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)(54,56)(55,60)(57,65)(58,67)(59,61)(62,68)(63,69)(64,66)(70,72)(71,77)(73,80)(74,81)(75,82)(76,79)(78,85)(83,84);(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)(55,59,62,63)(57,65)(58,67)(60,61,68,69)(64,66)(70,71,76,83,74,78,73,75)(72,77,79,84,81,85,80,82);(54,55)(56,63,65,59)(57,62)(58,61)(60,64,68,67)(66,69)(71,75)(72,79)(73,76)(78,83)(80,81)(82,84)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,d^2=gh,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,f^c=fh,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, 67143);