#27281885

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1289 ⋊ C₂
Sylow	C₅₃ ⋊ #9474

Subgroups

	subgroup	G/H
center	C₂²	#16810501
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1289	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂²×C₂²⋊D₅₃	D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1289	D₅₃
hypocenter	C₅₃	#9474
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1289

576 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 77 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)(55,60)(58,61)(62,65)(63,68)(64,70)(66,73)(67,74)(69,72)(71,76)(75,77);(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)(54,55,59,61,57,60,56,58)(62,64,66,71)(63,67,69,75)(65,70,73,76)(68,74,72,77);(55,58)(56,59)(60,61)(62,63)(64,67)(65,72,73,68)(66,69)(70,74,76,77)(71,75)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfgh,d^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^b=eh,i^b=i^52,d^c=dh,f^c=fh,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, 67361);