#27289234

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10690

Subgroups

	subgroup	G/H
center	C₂²	C₅₃⋊#401
commutator, prime radical, cosocle	C₂⁵×C₅₃	C₂³
Frattini	C₂⁵	C₂²×D₅₃
Fitting	C₅₃×#1304	C₂
socle	C₂²×C₅₃	#401
hypercenter	#1304	D₅₃
hypocenter	C₅₃	#10690
LCS	G ▷ C₂⁵×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #1304

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 1696

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

Polycyclic presentation:

2^8,53|a^2=def,b^2=de,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fg,i^b=i^52,e^c=eg,f^c=fh

Conjugacy classes & character table

701 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74710);