#27304423

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1411 ⋊ C₂
Sylow	C₅₃ ⋊ #13589

Subgroups

	subgroup	G/H
center	C₂	#353×D₅₃
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄⋊Q₈	C₂²×D₅₃
Fitting	C₅₃×#1411	C₂
socle	C₂×C₅₃	C₂×#353
prime radical	#20285144	C₂
cosocle	C₄⋊Q₈×C₅₃	C₂³
hypercenter	#1411	D₅₃
hypocenter	C₅₃	#13589
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #1411

408 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 59 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)(54,57,59,71)(55,61,64,62)(56,66,69,68)(58,70,74,73)(60,65,77,76)(63,78,81,79)(67,82,83,72)(75,80,85,84);(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)(54,56)(55,60)(57,66)(58,72)(59,69)(61,65)(62,76)(63,80)(64,77)(67,70)(68,71)(73,83)(74,82)(75,79)(78,85)(81,84);(54,55,58,63,59,64,74,81)(56,65,67,80,69,76,83,84)(57,61,70,78,71,62,73,79)(60,72,75,66,77,82,85,68)

Polycyclic presentation:

2^8,53|a^2=e,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dgh,e^b=efg,f^b=fh,i^b=i^52,i^c=i^52,e^d=egh,f^e=fh

Conjugacy classes & character table

476 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89899);