#27299847

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#360 ⋊ C₂
Sylow	C₅₃ ⋊ #12752

Subgroups

	subgroup	G/H
center	C₂	#20274644
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#360	C₂
socle	C₂×C₅₃	#1786
prime radical	#20289484	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#360	D₅₃
hypocenter	C₅₃	#12752
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

888 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85323);