#27302342

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#360 ⋊ C₂
Sylow	C₅₃ ⋊ #13211

Subgroups

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#360	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	#20285139	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#360	D₅₃
hypocenter	C₅₃	#13211
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

494 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:

(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)(59,65)(64,78)(66,80)(67,77)(68,81)(73,88)(79,89)(82,87);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(58,60,69,83,76,86,61,74,62,71,84,85,63,72,75,70)(59,64,73,87,77,89,66,81,65,78,88,82,67,79,80,68);(55,57)(58,59)(60,68,71,81)(61,73)(62,65)(63,77)(64,70,78,74)(66,69)(67,76)(72,87,86,82)(75,88)(79,83,89,85)(80,84)

Polycyclic presentation:

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

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 87818);