#27303209

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1389 ⋊ C₂
Sylow	C₅₃ ⋊ #13373

Subgroups

	subgroup	G/H
center	C₄	#16810968
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1389	C₂
socle	C₂×C₅₃	#2023
prime radical	C₂×#13133239	C₂²
hypercenter	#1389	D₅₃
hypocenter	C₅₃	#13373
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1389

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

normal subgroups only

This group has 364 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) = 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)(57,72)(58,69)(59,66)(60,67)(62,79)(63,80)(64,77)(65,81)(68,73)(70,82)(71,85)(74,84)(75,76)(78,83);(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)(56,66,67,69)(57,71)(61,76,77,80)(62,78)(65,79,82,83)(68,85,84,72)(70,81)(73,74);(54,55)(56,65,67,82)(57,70)(58,63)(59,75)(60,64)(61,68,77,84)(62,74)(66,83,69,79)(71,81)(72,80,85,76)(73,78)

Polycyclic presentation:

2^8,53|a^2=d,b^2=d,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^b=eh,i^b=i^52,d^c=dh,e^c=egh,f^c=fh,g^c=gh,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, 88685);