#27303273

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2139 ⋊ C₂
Sylow	C₅₃ ⋊ #13384

Subgroups

	subgroup	G/H
center	C₂²	#16810755
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2139	C₂
socle	C₂²×C₅₃	#447
prime radical	#20284381	C₂
hypercenter	#2139	D₅₃
hypocenter	C₅₃	#13384
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2139

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

normal subgroups only

This group has 408 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) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 89 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)(54,56,57,55)(59,65)(60,70)(61,74)(62,75)(64,77)(66,68)(67,81)(69,82)(71,85)(72,73)(76,87)(78,89)(79,88)(80,83);(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)(54,55)(56,57)(58,60)(59,64)(61,71)(62,70)(63,73)(65,80)(66,77)(67,83)(68,78)(69,79)(72,75)(74,84)(76,85)(81,88)(82,89)(86,87);(55,56)(58,59)(60,64)(61,65)(62,67)(63,69)(66,74)(68,75)(70,83)(71,80)(72,78)(73,79)(76,82)(77,84)(81,87)(85,89)(86,88)

Polycyclic presentation:

2^8,53|c^2=d,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,f^b=fh,i^b=i^52,e^c=eg,f^c=fh,g^c=gh

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, 88749);