#27484901

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2031 ⋊ C₂
Sylow	C₅₃ ⋊ #29233

Subgroups

	subgroup	G/H
center	C₄	#16811335
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2031	C₂
socle	C₂×C₅₃	#2572
prime radical	#16811421	C₂²
hypercenter	#2031	D₅₃
hypocenter	C₅₃	#29233
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2031

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 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)(55,63,65,64)(56,68)(61,66,77,81)(62,79,71,80)(67,83)(69,70)(72,84)(76,82,85,78);(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)(54,57)(55,62)(56,67)(58,73)(59,75)(60,74)(61,76)(63,79)(64,80)(65,71)(66,82)(68,83)(69,72)(70,84)(77,85)(78,81);(54,56)(55,61,65,77)(57,72,74,84)(58,69)(59,70)(60,68)(62,78)(63,81,64,66)(67,73,83,75)(71,82)(76,80)(79,85);(54,55,59,64,60,65,58,63)(56,66,70,77,68,81,69,61)(57,71,73,80,74,62,75,79)(67,78,72,85,83,82,84,76)

Polycyclic presentation:

2^8,53|a^2=g,b^2=eg,d^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=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, 270377);