#27537107

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2155 ⋊ C₂
Sylow	C₅₃ ⋊ #33404

Subgroups

	subgroup	G/H
center	C₂	C₂×#106×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2155	C₂
socle	C₂×C₅₃	C₂²×#106
prime radical	C₂×#11719945	D₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2155	D₅₃
hypocenter	C₅₃	#33404
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 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)(56,64)(61,73)(65,77)(66,78)(70,81)(74,83)(79,85)(82,84);(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)(57,65)(58,66)(59,64)(60,70)(62,73)(63,74)(67,79)(68,77)(69,78)(71,81)(72,82)(75,83)(76,84)(80,85);(54,55,57,60,58,63,67,72,59,62,68,71,69,75,80,76)(56,61,65,70,66,74,79,82,64,73,77,81,78,83,85,84);(55,60,62,71)(57,67)(58,69)(61,70,73,81)(63,76,75,72)(65,79)(66,78)(68,80)(74,84,83,82)(77,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=e,e^2=g,g^2=h,b^a=be,e^a=eg,g^a=gh,f^b=fh,d^c=dh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322583);