#27353440

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1569 ⋊ C₂
Sylow	C₅₃ ⋊ #17841

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1569	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719945	Q₈
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1569	D₅₃
hypocenter	C₅₃	#17841
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1569

592 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)(56,62)(59,65)(63,68)(66,69)(70,73,75,74)(71,77,79,78)(72,82,81,83)(76,85,84,80);(54,56)(55,59)(57,63)(58,62)(60,66)(61,65)(64,68)(67,69)(72,81)(76,84)(80,85)(82,83);(70,72,74,83,75,81,73,82)(71,76,77,85,79,84,78,80);(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)(54,55,57,60,58,61,64,67)(56,59,63,66,62,65,68,69)(70,71,74,78,75,79,73,77)(72,80,83,84,81,85,82,76)

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 138916);