#27297423

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1342 ⋊ C₂
Sylow	C₅₃ ⋊ #12299

Subgroups

	subgroup	G/H
center	C₂	#20273736
commutator, cosocle	C₂×M₁₆×C₅₃	C₂³
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₅₃×#1342	C₂
socle	C₂×C₅₃	#1613
prime radical	C₂³×D₅₃	SD₁₆
hypercenter	#1342	D₅₃
hypocenter	C₅₃	#12299
LCS	G ▷ C₂×M₁₆×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1342

324 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 1696

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

(8,25)(11,27)(15,31)(24,28)(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)(77,78)(79,80)(81,82)(83,84);(1,3,6,13)(2,7,10,23)(4,16,17,29)(5,20,21,14)(8,24,25,28)(9,26,19,22)(11,15,27,31)(12,18,30,32);(1,2,5,19,6,10,21,9)(3,11,14,24,13,27,20,28)(4,15,18,8,17,31,32,25)(7,16,26,12,23,29,22,30)(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)(78,79)(80,81)(82,83)(84,85)

Polycyclic presentation:

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

Conjugacy classes & character table

545 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 82899);