#27287333

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1284 ⋊ C₂
Sylow	C₅₃ ⋊ #10388

Subgroups

	subgroup	G/H
center	C₂²	#16810536
commutator, cosocle	C₂³×C₄×C₅₃	C₂³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1284	C₂
socle	C₂²×C₅₃	#393
prime radical	C₂²×#11719945	C₂²
hypercenter	#1284	D₅₃
hypocenter	C₅₃	#10388
LCS	G ▷ C₂³×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1284

594 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 73 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)(58,61)(62,64)(63,68)(65,72)(66,74)(67,73)(69,77)(70,76)(71,75);(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)(55,58)(57,60)(59,61)(64,72,73,74)(68,71,76,77);(54,55)(56,58)(57,59)(60,61)(62,63,67,70)(64,71)(65,69,66,75)(68,72)(73,77)(74,76)

Polycyclic presentation:

2^8,53|a^2=dg,b^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eh,i^b=i^52,d^c=dg,e^c=eg,f^c=fg,i^c=i^52

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 72809);