#27402618

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1861 ⋊ C₂
Sylow	C₅₃ ⋊ #22695

Subgroups

	subgroup	G/H
center	C₄	#16811255
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1861	C₂
socle	C₂×C₅₃	D₄×C₄⋊C₄
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1861	D₅₃
hypocenter	C₅₃	#22695
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1861

808 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)(55,64)(56,69)(57,70)(59,78)(60,77)(61,74)(62,73)(63,81)(65,85)(66,84)(67,79)(68,83)(71,76)(72,75)(80,82);(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)(54,57,60,75,61,76,59,69)(55,63,66,73,67,83,65,82)(56,58,70,77,72,74,71,78)(62,79,68,85,80,64,81,84);(54,56)(55,62)(57,74)(58,76)(59,71)(60,70)(61,72)(63,64)(65,81)(66,68)(67,80)(69,77)(73,84)(75,78)(79,83)(82,85);(54,55,59,66,61,67,60,65)(56,68,71,80,72,81,70,62)(57,73,69,83,76,82,75,63)(58,64,78,84,74,79,77,85)

Polycyclic presentation:

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

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 188094);