#27358447

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #18345

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1616	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₅₃×#537	C₂²
hypercenter	#1616	D₅₃
hypocenter	C₅₃	#18345
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1616

888 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)(54,57,59,58)(55,62,63,61)(56,65,66,67)(60,64,69,68);(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)(54,57,59,58)(55,61,63,62)(56,65,66,67)(60,68,69,64)(70,72,74,79)(71,76,77,85)(73,82,80,78)(75,83,81,84);(54,56)(55,60)(57,65)(58,67)(59,66)(61,68)(62,64)(63,69)(70,71)(72,76)(73,81)(74,77)(75,80)(78,83)(79,85)(82,84);(54,55,59,63)(56,64,66,68)(57,61,58,62)(60,67,69,65)(71,75)(73,80)(76,84)(77,81)(78,82)(83,85)

Polycyclic presentation:

2^8,53|a^2=gh,c^2=g,d^2=gh,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,e^c=eh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

635 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 143923);