#27278488

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1569 ⋊ C₂
Sylow	C₅₃ ⋊ #8917

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133185
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1569	C₂
socle	C₂²×C₅₃	#385
prime radical	#13133191	Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1569	D₅₃
hypocenter	C₅₃	#8917
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1569

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

normal subgroups only

This group has 338 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) = 6784

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

(1,2)(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)(54,56)(55,59)(57,67)(58,63)(60,69)(61,68)(62,65)(64,66)(70,73)(71,78)(72,82)(74,85)(75,81)(76,77)(79,84)(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)(57,66)(60,65)(62,69)(64,67)(70,72,75,78,76,84,74,83)(71,77,79,85,80,73,82,81);(54,55,58,61)(56,62,63,69)(57,65,66,60)(59,67,68,64)(70,71,75,79,76,80,74,82)(72,81,78,77,84,85,83,73)

Polycyclic presentation:

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

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, 63964);