#27302826

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#361 ⋊ C₂
Sylow	C₅₃ ⋊ #13310

Subgroups

	subgroup	G/H
center	C₂	#20275712
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#361	C₂
socle	C₂×C₅₃	#1999
hypercenter	C₂×#361	D₅₃
hypocenter	C₅₃	#13310
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

552 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 93 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:

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

Polycyclic presentation:

2^8,53|d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,i^a=i^52,c^b=cf,d^b=dg,e^b=eh,g^b=gh,e^c=eh,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, 88302);