#27303243

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1394 ⋊ C₂
Sylow	C₅₃ ⋊ #13379

Subgroups

	subgroup	G/H
center	C₂²	#16810811
commutator, cosocle	C₂²×Q₈×C₅₃	C₂³
Frattini	C₂²×Q₈	C₂²×D₅₃
Fitting	C₅₃×#1394	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
prime radical	C₂×#13133265	C₂²
hypercenter	#1394	D₅₃
hypocenter	C₅₃	#13379
LCS	G ▷ C₂²×Q₈×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #1394

540 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 73 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:

(9,11)(10,15)(12,18)(13,21)(14,23)(16,24)(17,22)(19,20)(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)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,17,18,24)(14,22,20,16)(15,23,21,19);(1,2,4,6)(3,7,8,5)(10,13)(11,15)(14,20)(17,19)(18,21)(23,24)(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)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)

Polycyclic presentation:

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

Conjugacy classes & character table

707 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88719);