#27280916

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1291 ⋊ C₂
Sylow	C₅₃ ⋊ #9313

Subgroups

	subgroup	G/H
center	C₂²	#16810501
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1291	C₂
socle	C₂²×C₅₃	#386
prime radical	#20284867	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1291	D₅₃
hypocenter	C₅₃	#9313
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1291

428 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 89 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)(56,64)(60,68)(63,69)(65,66)(70,73)(71,77)(72,80)(74,84)(75,82)(76,83)(78,85)(79,81);(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)(55,60,62,68)(57,65)(58,66)(59,64)(61,63,67,69)(70,72,75,78,76,79,74,71)(73,81,82,77,83,80,84,85);(54,55,58,61,59,62,57,67)(56,63,66,68,64,69,65,60)(70,71,75,79,76,78,74,72)(73,80,82,77,83,81,84,85)

Polycyclic presentation:

2^8,53|a^2=f,b^2=df,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,f^b=fh,i^b=i^52,e^c=eh,i^c=i^52

Conjugacy classes & character table

869 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 66392);