#27537547

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#360 ⋊ C₂
Sylow	C₅₃ ⋊ #33469

Subgroups

	subgroup	G/H
center	C₂	C₂×#16810732
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#360	C₂
socle	C₂×C₅₃	C₂×#442
prime radical	#20287849	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#360	D₅₃
hypocenter	C₅₃	#33469
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(56,65)(61,78)(64,82)(66,69)(74,79)(77,85)(80,81)(83,84);(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)(54,56,58,64,59,65,71,82)(55,61,72,81,62,78,63,80)(57,69,67,84,70,66,73,83)(60,74,68,85,75,79,76,77);(54,55)(56,61)(57,68)(58,72)(59,62)(60,73)(63,71)(64,81)(65,78)(66,77)(67,75)(69,85)(70,76)(74,83)(79,84)(80,82);(55,60,62,75)(56,64)(57,67)(58,71)(61,77,78,85)(63,68,72,76)(65,82)(66,69)(70,73)(74,81,79,80)

Polycyclic presentation:

2^8,53|a^2=f,c^2=fg,e^2=g,g^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,d^c=dh,e^c=eh,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, 323023);