#27281636

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9427

Subgroups

	subgroup	G/H
center	C₂²	#16810499
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1515	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	#120
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1515	D₅₃
hypocenter	C₅₃	#9427
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1515

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

normal subgroups only

This group has 422 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) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 67112);