#27285467

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10093

Subgroups

	subgroup	G/H
center	C₂²	#16810531
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#3234	C₂
socle	C₂²×C₅₃	#392
prime radical	C₅₃×#532	C₂²
hypercenter	#3234	D₅₃
hypocenter	C₅₃	#10093
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3234

496 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 85 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)(54,56,59,66)(55,61,63,69)(57,65,58,67)(60,68,62,64)(70,73,75,72)(71,77,79,80)(74,81,84,82)(76,78,85,83);(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)(55,60)(57,58)(61,64)(62,63)(65,67)(68,69)(70,72,75,73)(71,76,79,85)(74,82,84,81)(77,83,80,78);(54,55)(56,64)(57,60)(58,62)(59,63)(61,65)(66,68)(67,69)(70,71)(72,80)(73,77)(74,83)(75,79)(76,81)(78,84)(82,85)

Polycyclic presentation:

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

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, 70943);