#27288902

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10635

Subgroups

	subgroup	G/H
center	C₂²	#16810548
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1960	C₂
socle	C₂²×C₅₃	#397
prime radical	C₂³×C₅₃	#124
hypercenter	#1960	D₅₃
hypocenter	C₅₃	#10635
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 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)(56,58)(57,61)(59,60)(62,65,66,81,68,83,67,82)(63,70,71,79,73,88,72,89)(64,75,76,91,78,92,77,80)(69,84,74,87,86,90,85,93);(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)(55,59)(57,60)(58,61)(62,64,68,78)(63,69,73,86)(65,80,83,91)(66,77,67,76)(70,87,88,93)(71,85,72,74)(75,81,92,82)(79,84,89,90);(55,58)(59,61)(62,63,66,71,68,73,67,72)(64,74,76,86,78,85,77,69)(65,79,82,70,83,89,81,88)(75,90,80,87,92,84,91,93)

Polycyclic presentation:

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

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74378);