#27350294

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1470 ⋊ C₂
Sylow	C₅₃ ⋊ #17413

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1470	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1470	D₅₃
hypocenter	C₅₃	#17413
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1470

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,d^2=g,f^2=h,b^a=be,e^a=eg,f^b=fg,i^b=i^52,d^c=dh,i^d=i^52

Conjugacy classes & character table

887 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 135770);