#27383777

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1816 ⋊ C₂
Sylow	C₅₃ ⋊ #20678

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1816	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#41×D₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1816	D₅₃
hypocenter	C₅₃	#20678
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1816

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(58,62,64,63)(59,67,69,68)(60,72,74,73)(61,76,78,77)(65,80,81,70)(66,82,83,75)(71,86,88,87)(79,89,85,84);(58,61)(59,66)(60,71)(62,76)(63,77)(64,78)(65,79)(67,82)(68,75)(69,83)(70,84)(72,86)(73,87)(74,88)(80,89)(81,85);(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)(58,60,62,72,64,74,63,73)(59,65,68,70,69,81,67,80)(61,71,76,86,78,88,77,87)(66,79,75,84,83,85,82,89);(54,55,56,57)(58,59,63,68,64,69,62,67)(60,70,73,81,74,80,72,65)(61,75,76,66,78,82,77,83)(71,85,86,84,88,79,87,89)

Polycyclic presentation:

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

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 169253);