#27540581

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

Aliases

No aliases for this group.

Splittings

Fitting	D₄×Q₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ #33832

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	D₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20286402	C₂
hypercenter	D₄×Q₁₆	D₅₃
hypocenter	C₅₃	#33832
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

686 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)(59,65)(60,70)(61,75)(62,76)(64,80)(67,68)(69,82)(71,72)(73,88)(74,83)(77,84)(78,79)(81,85)(87,89);(55,57);(58,60,63,73)(59,64,68,82)(61,70,77,88)(62,78,76,72)(65,81,67,87)(66,85,86,89)(69,83,80,74)(71,84,79,75);(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)(56,57)(58,59,63,68)(60,69,73,80)(61,74,77,83)(62,65,76,67)(64,79,82,71)(66,84,86,75)(70,87,88,81)(72,89,78,85)

Polycyclic presentation:

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

Conjugacy classes & character table

953 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 326057);