#27279207

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1282 ⋊ C₂
Sylow	C₅₃ ⋊ #9038

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1282	C₂
socle	C₂²×C₅₃	#386
prime radical	#16811134	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1282	D₅₃
hypocenter	C₅₃	#9038
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1282

468 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 93 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)(55,60)(56,62)(58,65)(59,66)(61,64)(63,68)(67,69)(70,72)(71,77)(73,82)(74,80)(75,83)(76,79)(78,85)(81,84);(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)(55,59,61,67)(57,62,65,68)(60,66,64,69)(70,71,76,75,74,78,73,81)(72,77,79,83,80,85,82,84);(54,55)(56,59)(57,64,65,60)(58,61)(62,66,68,69)(63,67)(71,75,78,81)(72,79,80,82)

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 64683);