#27303629

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13439

Subgroups

	subgroup	G/H
center	C₂²	#16810856
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₂²×D₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#466
hypercenter	C₂²×D₁₆	D₅₃
hypocenter	C₅₃	#13439
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 95 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)(55,57)(58,60,61,72,76,86,78,89,64,74,62,79,63,73,75,87)(59,66,67,71,80,85,77,88,70,83,68,84,69,82,65,81);(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)(59,65)(60,72)(61,75)(62,78)(63,76)(67,69)(68,80)(70,77)(71,81)(73,89)(74,79)(82,85)(84,88)(86,87);(54,55)(56,57)(58,59)(60,71)(61,65)(62,77)(63,80)(64,70)(66,72)(67,75)(68,78)(69,76)(73,88)(74,84)(79,83)(81,86)(82,89)(85,87)

Polycyclic presentation:

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

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89105);