#27286360

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1518 ⋊ C₂
Sylow	C₅₃ ⋊ #10223

Subgroups

	subgroup	G/H
center	C₂²	#16810532
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1518	C₂
socle	C₂²×C₅₃	#392
prime radical	#16811173	C₂²
hypercenter	#1518	D₅₃
hypocenter	C₅₃	#10223
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1518

484 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 95 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 71836);