#27301381

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1391 ⋊ C₂
Sylow	C₅₃ ⋊ #13045

Subgroups

	subgroup	G/H
center	C₄	#16810811
commutator	C₂×D₄×C₅₃	C₂²×C₄
Frattini	C₂×#41	C₂²×D₅₃
Fitting	C₅₃×#1391	C₂
socle	C₂×C₅₃	#1889
prime radical	#20284109	C₂
cosocle	C₂×#41×C₅₃	C₂³
hypercenter	#1391	D₅₃
hypocenter	C₅₃	#13045
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #1391

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

normal subgroups only

This group has 654 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) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

(2,9)(8,23)(10,25)(11,18)(12,27)(22,26)(24,32)(28,29)(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)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84);(1,3,13,21,7,16,20,30)(2,8,22,28,11,24,25,27)(4,14,6,15,17,31,5,19)(9,23,26,29,18,32,10,12);(1,2,6,9,7,11,5,18)(3,12,15,28,16,29,19,27)(4,10,13,22,17,26,20,25)(8,21,23,31,24,30,32,14)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 86857);