#27293531

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1328 ⋊ C₂
Sylow	C₅₃ ⋊ #11523

Subgroups

	subgroup	G/H
center	C₄	#16810595
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₅₃×#1328	C₂
socle	C₂×C₅₃	#1464
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	#1328	D₅₃
hypocenter	C₅₃	#11523
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #1328

280 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 69 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79007);