#27304876

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1415 ⋊ C₂
Sylow	C₅₃ ⋊ #13686

Subgroups

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1415	C₂
socle	C₂²×C₅₃	C₂×D₁₆
prime radical	#20276963	C₂
hypercenter	#1415	D₅₃
hypocenter	C₅₃	#13686
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1415

220 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|b^2=d,c^2=h,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,i^b=i^52,e^c=ef,f^c=fg,g^c=gh,i^c=i^52

Conjugacy classes & character table

1037 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90352);