#27281617

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1536 ⋊ C₂
Sylow	C₅₃ ⋊ #9423

Subgroups

	subgroup	G/H
center	C₂²	#16810500
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1536	C₂
socle	C₂²×C₅₃	#386
prime radical	#20285168	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1536	D₅₃
hypocenter	C₅₃	#9423
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

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

normal subgroups only

This group has 540 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) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 77 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)(57,59)(60,61)(62,65)(63,69)(64,72)(66,75)(67,73)(68,74)(70,76)(71,77);(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)(54,56,58,55)(57,60,59,61)(62,64,67,70,68,71,66,63)(65,72,73,76,74,77,75,69);(54,55,59,61,58,56,57,60)(62,63,67,71,68,70,66,64)(65,72,73,69,74,77,75,76)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfgh,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,i^b=i^52,d^c=dgh,e^c=eg,f^c=fgh

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 67093);