#27287235

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10373

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133204
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1995	C₂
socle	C₂²×C₅₃	#393
prime radical	C₂³×C₅₃	#120
hypercenter	#1995	D₅₃
hypocenter	C₅₃	#10373
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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)(54,56,57,63)(55,60,59,68)(58,67,65,64)(61,69,66,62)(70,72)(71,76)(73,82)(74,79)(75,81)(77,85)(78,84)(80,83);(55,59)(60,68)(61,66)(62,69)(70,71)(72,80,81,85)(73,83)(74,77)(75,78)(76,82,84,79);(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)(54,55,57,59)(56,62,63,69)(58,66,65,61)(60,67,68,64)(72,79,81,82)(76,80,84,85)

Polycyclic presentation:

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

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, 72711);