#27282105

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1305 ⋊ C₂
Sylow	C₅₃ ⋊ #9518

Subgroups

	subgroup	G/H
center	C₂²	#16810501
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1305	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂²×C₂²⋊D₅₃	C₂×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1305	D₅₃
hypocenter	C₅₃	#9518
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1305

518 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 67581);