#27537135

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2159 ⋊ C₂
Sylow	C₅₃ ⋊ #33410

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2159	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#11719945	D₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2159	D₅₃
hypocenter	C₅₃	#33410
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 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)(60,69)(66,80)(70,83)(71,84)(74,85)(78,88)(81,86)(87,89);(58,60)(59,66)(61,70)(62,71)(63,69)(64,74)(65,78)(67,81)(68,80)(72,84)(73,83)(75,86)(76,87)(77,85)(79,88)(82,89);(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)(58,59,65,76,62,67,73,77,63,68,79,82,72,75,61,64)(60,66,78,87,71,81,83,85,69,80,88,89,84,86,70,74);(54,55,56,57)(59,64)(61,65)(62,72)(66,74)(67,82)(68,77)(70,78)(71,84)(73,79)(75,76)(80,85)(81,89)(83,88)(86,87)

Polycyclic presentation:

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

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322611);