#27277130

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #8641

Subgroups

	subgroup	G/H
center	C₂²	#16810482
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1309	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1309	D₅₃
hypocenter	C₅₃	#8641
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

336 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 101 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)(54,56,57,63)(55,60,59,62)(58,66,65,64)(61,69,67,68)(70,72,74,81)(71,77,78,85)(73,80,76,79)(75,82,83,84);(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)(55,59)(60,62)(61,67)(68,69)(70,71,76,83,74,78,73,75)(72,77,79,84,81,85,80,82);(54,55,58,61,57,59,65,67)(56,62,64,69,63,60,66,68)(71,75)(72,79)(73,76)(78,83)(80,81)(82,84)

Polycyclic presentation:

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

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, 62606);