#27290913

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#421 ⋊ C₂
Sylow	C₅₃ ⋊ #11026

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133189
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#421	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂²×#13133237	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#421	D₅₃
hypocenter	C₅₃	#11026
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#421

1278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(58,61)(59,65)(60,69)(62,77)(63,75)(64,79)(66,72)(67,85)(68,86)(70,88)(71,87)(73,76)(74,78)(80,89)(81,84)(82,83);(55,57)(58,60)(59,64)(61,73)(62,76)(63,71)(65,81)(66,68)(67,84)(69,74)(70,78)(72,89)(75,88)(77,87)(79,82)(80,83)(85,86);(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)(56,57)(58,59)(60,68,71,64)(61,72,75,65)(62,67)(63,66)(69,79)(70,84,76,80)(73,81)(74,85,77,82)(78,83)(86,87)(88,89)

Polycyclic presentation:

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

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76389);