#27362119

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1624 ⋊ C₂
Sylow	C₅₃ ⋊ #18706

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1624	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#1624	D₅₃
hypocenter	C₅₃	#18706
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1624

908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|b^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh,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, 147595);