#27422340

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3056 ⋊ C₂
Sylow	C₅₃ ⋊ #24470

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3056	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288387	C₂
hypercenter	#3056	D₅₃
hypocenter	C₅₃	#24470
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #3056

1360 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)(59,68)(60,63)(61,75)(64,73)(65,78)(66,79)(67,83)(69,82)(70,71)(72,76)(74,81)(77,88)(80,89)(84,87)(85,86);(55,57)(58,61,65,77)(59,67,70,85)(60,72,73,89)(62,75,78,88)(63,76,64,80)(66,81,82,87)(68,83,71,86)(69,84,79,74);(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)(58,60)(59,66)(61,72)(62,64)(63,78)(65,73)(67,81)(68,69)(70,82)(71,79)(74,86)(75,80)(76,88)(77,89)(83,84)(85,87);(58,59,64,69,65,70,63,79)(60,71,62,66,73,68,78,82)(61,74,76,85,77,84,80,67)(72,87,88,83,89,81,75,86)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,i^b=i^52,f^c=fh,e^d=eh,i^d=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, 207816);