#27538059

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2194 ⋊ C₂
Sylow	C₅₃ ⋊ #33539

Subgroups

	subgroup	G/H
center	C₂	#447×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2194	C₂
socle	C₂×C₅₃	C₂×#447
hypercenter	#2194	D₅₃
hypocenter	C₅₃	#33539
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2194

1130 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)(55,62)(60,76)(61,79)(63,81)(64,75)(65,80)(70,84)(77,78);(54,56)(55,61)(57,66)(58,68)(59,69)(60,63)(62,79)(64,80)(65,75)(67,72)(70,78)(71,73)(74,83)(76,81)(77,84)(82,85);(54,56)(55,60)(57,71)(58,67)(59,69)(61,63)(62,76)(64,77)(65,70)(66,73)(68,72)(74,82)(75,78)(79,81)(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)(54,55)(56,65)(57,70)(58,75)(59,62)(60,66)(61,67)(63,73)(64,72)(68,79)(69,80)(71,77)(74,84)(76,83)(78,82)(81,85)

Polycyclic presentation:

2^8,53|e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gh,i^a=i^52,e^b=eg,g^b=gh,f^c=fh,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, 323535);