#27431385

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2063 ⋊ C₂
Sylow	C₅₃ ⋊ #25143

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2063	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20283959	C₂
hypercenter	#2063	D₅₃
hypocenter	C₅₃	#25143
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2063

996 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)(55,57)(58,62,64,63)(59,67,69,68)(60,71,73,72)(61,76,77,78)(65,80,82,81)(66,74,84,85)(70,87,75,88)(79,86,83,89);(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)(58,61,64,77)(59,66,69,84)(60,70,73,75)(62,76,63,78)(65,79,82,83)(67,74,68,85)(71,87,72,88)(80,86,81,89);(55,57)(58,60,63,72,64,73,62,71)(59,65,68,81,69,82,67,80)(61,75,76,88,77,70,78,87)(66,83,74,89,84,79,85,86);(54,55)(56,57)(58,59,63,68,64,69,62,67)(60,65,72,81,73,82,71,80)(61,74,76,84,77,85,78,66)(70,86,87,83,75,89,88,79)

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 216861);