#27536900

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2129 ⋊ C₂
Sylow	C₅₃ ⋊ #33360

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2129	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	#20288497	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	#2129	D₅₃
hypocenter	C₅₃	#33360
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2129

1188 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)(54,57,59,70)(55,62,64,79)(56,65,67,68)(58,69,71,83)(60,73,63,78)(61,74,76,77)(66,80,81,82)(72,84,75,85);(54,56,59,67)(55,61,64,76)(57,68,70,65)(58,66,71,81)(60,72,63,75)(62,77,79,74)(69,82,83,80)(73,85,78,84);(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)(55,60)(58,71)(61,72)(62,73)(63,64)(66,81)(69,83)(74,84)(75,76)(77,85)(78,79)(80,82);(54,55)(56,61)(57,62)(58,63,71,60)(59,64)(65,74)(66,75,81,72)(67,76)(68,77)(69,78,83,73)(70,79)(80,85,82,84)

Polycyclic presentation:

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

Conjugacy classes & character table

644 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322376);