#27301095

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1387 ⋊ C₂
Sylow	C₅₃ ⋊ #13003

Subgroups

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1387	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×D₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1387	D₅₃
hypocenter	C₅₃	#13003
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 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);(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)(58,60,63,80,64,74,79,73)(59,65,81,83,69,84,68,87)(61,72,76,71,77,89,62,78)(66,85,86,88,75,82,67,70);(54,55)(56,57)(58,59,62,67,79,81,77,66,64,69,76,86,63,68,61,75)(60,70,72,65,80,82,71,87,74,88,89,84,73,85,78,83)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfg,d^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,f^b=fg,g^b=gh,i^b=i^52,i^c=i^52

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 86571);