#27287056

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10339

Subgroups

	subgroup	G/H
center	C₂²	#16810531
commutator, prime radical, cosocle	C₂⁵×C₅₃	C₂³
Frattini	C₂⁵	C₂²×D₅₃
Fitting	C₅₃×#2483	C₂
socle	C₂²×C₅₃	#392
hypercenter	#2483	D₅₃
hypocenter	C₅₃	#10339
LCS	G ▷ C₂⁵×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #2483

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

normal subgroups only

This group has 738 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) = 1696

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=g,c^2=h,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eh,d^c=dg,i^c=i^52

Conjugacy classes & character table

1022 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 72532);