#27284975

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2633 ⋊ C₂
Sylow	C₅₃ ⋊ #10019

Subgroups

	subgroup	G/H
center	C₂²	#16810531
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2633	C₂
socle	C₂²×C₅₃	#392
prime radical	#20284806	C₂
hypercenter	#2633	D₅₃
hypocenter	C₅₃	#10019
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2633

540 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 81 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:

(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)(57,66)(58,63)(60,69)(61,68)(62,65)(64,67)(70,72,75,82)(71,77,79,85)(73,81,74,83)(76,84,78,80);(59,68)(62,69)(71,76)(73,74)(77,80)(78,79)(81,83)(84,85);(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)(56,62)(57,65)(58,61)(59,64)(60,67)(63,69)(66,68)(70,71)(72,80)(73,76)(74,78)(75,79)(77,81)(82,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

863 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 70451);