#27469905

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2081 ⋊ C₂
Sylow	C₅₃ ⋊ #27987

Subgroups

	subgroup	G/H
center	C₂	#20279202
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2081	C₂
socle	C₂×C₅₃	#2504
prime radical	#16811190	C₂²
hypercenter	#2081	D₅₃
hypocenter	C₅₃	#27987
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2081

638 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)(54,58,59,77,61,79,60,78)(55,64,65,68,66,84,80,72)(56,63,69,71,70,83,81,85)(57,73,74,62,76,67,75,82);(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)(55,63,66,83)(56,68,70,72)(58,73)(59,74)(60,75)(61,76)(62,77)(64,69,84,81)(65,71,80,85)(67,79)(78,82);(54,56,59,69,61,70,60,81)(55,62,65,67,66,82,80,73)(57,72,74,64,76,68,75,84)(58,63,77,71,79,83,78,85);(54,55,59,80,61,66,60,65)(56,67,69,62,70,73,81,82)(57,71,75,83,76,85,74,63)(58,68,77,64,79,72,78,84)

Polycyclic presentation:

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 255381);