#27298341

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1357 ⋊ C₂
Sylow	C₅₃ ⋊ #12438

Subgroups

	subgroup	G/H
center	C₂	#20274069
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1357	C₂
socle	C₂×C₅₃	#1667
prime radical	#20276788	C₂
hypercenter	#1357	D₅₃
hypocenter	C₅₃	#12438
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1357

224 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 55 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)(55,62)(57,66)(58,67)(59,69)(60,73)(61,78)(63,80)(64,82)(65,70)(68,71)(72,83)(74,81)(75,79)(76,85)(77,84);(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)(54,55,61,77,58,63,57,60,59,64,70,75,71,74,72,76)(56,62,78,84,67,80,66,73,69,82,65,79,68,81,83,85);(55,60)(56,65)(57,70)(58,71)(61,72)(63,77)(64,76)(66,67)(68,83)(69,78)(73,79)(74,75)(80,81)(84,85)

Polycyclic presentation:

2^8,53|b^2=dfh,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,i^b=i^52,e^c=eg,g^c=gh,i^c=i^52

Conjugacy classes & character table

635 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 83817);