#27540592

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2217 ⋊ C₂
Sylow	C₅₃ ⋊ #33833

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2217	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2217	D₅₃
hypocenter	C₅₃	#33833
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2217

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|c^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gh,d^b=de,e^b=eg,g^b=gh,i^b=i^52,e^d=eg,g^d=gh,i^d=i^52

Conjugacy classes & character table

1031 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326068);