#27277587

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1285 ⋊ C₂
Sylow	C₅₃ ⋊ #8745

Subgroups

	subgroup	G/H
center	C₂²	#16810485
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1285	C₂
socle	C₂²×C₅₃	#381
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1285	D₅₃
hypocenter	C₅₃	#8745
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1285

384 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 85 points:

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=h,c^2=dh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,i^b=i^52,f^c=fgh

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 63063);