#27277305

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1286 ⋊ C₂
Sylow	C₅₃ ⋊ #8676

Subgroups

	subgroup	G/H
center	C₂²	#16810482
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1286	C₂
socle	C₂²×C₅₃	#380
prime radical	#16811173	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1286	D₅₃
hypocenter	C₅₃	#8676
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1286

434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 93 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)(55,58)(56,61)(59,60)(62,65)(63,69)(64,74)(66,78)(67,79)(68,81)(70,83)(71,77)(72,84)(73,87)(75,89)(76,90)(80,92)(82,91)(85,88)(86,93);(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)(62,64,73,86,67,76,66,75)(63,68,80,85,71,82,70,72)(65,74,87,93,79,90,78,89)(69,81,92,88,77,91,83,84);(54,55)(56,58)(57,60)(59,61)(62,63)(64,72)(65,77,79,69)(66,80)(67,71)(68,75)(70,73)(74,88,90,84)(76,85)(78,83,87,92)(81,89,91,93)(82,86)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 62781);