#27435507

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2046 ⋊ C₂
Sylow	C₅₃ ⋊ #25437

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2046	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#13133243	C₂²
hypercenter	#2046	D₅₃
hypocenter	C₅₃	#25437
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2046

894 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)(58,61,62,76,63,77,78,83)(59,66,67,74,69,86,68,85)(60,70,71,88,73,75,72,89)(64,79,65,81,80,87,82,84);(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)(54,55)(56,57)(58,60)(59,65)(61,75)(62,71)(63,73)(64,68)(66,84)(67,80)(69,82)(70,77)(72,78)(74,79)(76,89)(81,86)(83,88)(85,87);(54,55,56,57)(58,60,62,71,63,73,78,72)(59,64,67,65,69,80,68,82)(61,70,76,88,77,75,83,89)(66,79,74,81,86,87,85,84);(54,55,56,57)(58,59,63,69)(60,65,73,82)(61,74,77,85)(62,68,78,67)(64,72,80,71)(66,83,86,76)(70,87,75,79)(81,89,84,88)

Polycyclic presentation:

2^8,53|a^2=fh,b^2=ef,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 220983);