#27299368

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #12648

Subgroups

	subgroup	G/H
center	C₂²	#16810734
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1383	C₂
socle	C₂²×C₅₃	#442
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1383	D₅₃
hypocenter	C₅₃	#12648
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

422 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 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)(60,71,75,74)(61,63)(65,82,85,84)(66,68)(70,86,83,89)(72,88,73,87)(76,79)(78,80);(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)(54,55,56,57)(58,60,63,73,64,74,80,87,62,75,61,72,77,71,78,88)(59,65,68,83,69,84,79,89,67,85,66,70,81,82,76,86);(58,59,62,67)(60,70)(61,76,63,79)(64,81,77,69)(65,73)(66,80,68,78)(71,86)(72,85)(74,89)(75,83)(82,87)(84,88)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84844);