#27288780

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10613

Subgroups

	subgroup	G/H
center	C₂²	#16810545
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1574	C₂
socle	C₂²×C₅₃	#396
prime radical	C₂³×C₅₃	#126
hypercenter	#1574	D₅₃
hypocenter	C₅₃	#10613
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1574

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 85
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2)(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)(55,61)(56,65)(58,59)(62,63)(66,67)(68,69)(70,72,74,82)(71,77,80,84)(73,83,75,81)(76,79,85,78);(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)(54,56,57,66)(55,60,62,64)(58,65,59,67)(61,69,63,68)(70,72,73,81,74,82,75,83)(71,76,78,77,80,85,79,84);(54,55,59,61,57,62,58,63)(56,64,67,68,66,60,65,69)(70,71,75,78,74,80,73,79)(72,77,83,85,82,84,81,76)

Polycyclic presentation:

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

Conjugacy classes & character table

1022 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 74256);