#27352322

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1525 ⋊ C₂
Sylow	C₅₃ ⋊ #17687

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1525	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#20288635	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1525	D₅₃
hypocenter	C₅₃	#17687
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1525

896 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)(55,61,62,63)(56,65,66,67)(60,68,64,69)(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)(54,57,59,58)(55,61,62,63)(56,65,66,67)(60,68,64,69)(70,72)(71,76)(73,79)(74,78)(75,82)(77,83)(80,85)(81,84);(54,56)(55,60)(57,65)(58,67)(59,66)(61,68)(62,64)(63,69)(70,71)(72,76)(73,80)(74,77)(75,81)(78,83)(79,85)(82,84);(54,55,58,63,59,62,57,61)(56,64,65,69,66,60,67,68)(71,75,77,80)(76,82,83,85)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,d^2=g,e^2=h,f^2=g,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 137798);