#27411097

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2008 ⋊ C₂
Sylow	C₅₃ ⋊ #23412

Subgroups

	subgroup	G/H
center	C₂	#392×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2008	C₂
socle	C₂×C₅₃	C₂×#392
hypercenter	#2008	D₅₃
hypocenter	C₅₃	#23412
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2008

1948 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:

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

Polycyclic presentation:

2^8,53|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,e^c=eh,f^c=fh,i^d=i^52

Conjugacy classes & character table

644 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196573);