#27420010

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2010 ⋊ C₂
Sylow	C₅₃ ⋊ #24286

Subgroups

	subgroup	G/H
center	C₄	#16811186
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2010	C₂
socle	C₂×C₅₃	#2408
hypercenter	#2010	D₅₃
hypocenter	C₅₃	#24286
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2010

1322 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)(54,57,58,70,59,69,72,83)(55,61,62,68,63,75,71,82)(56,65,66,80,67,79,81,85)(60,74,64,77,73,78,76,84);(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)(57,69)(60,73)(61,75)(64,76)(65,79)(68,82)(70,83)(80,85);(54,56,58,66,59,67,72,81)(55,60,62,64,63,73,71,76)(57,65,70,80,69,79,83,85)(61,74,68,77,75,78,82,84);(54,55)(56,64)(57,68)(58,71)(59,63)(60,66)(61,70)(62,72)(65,78)(67,76)(69,82)(73,81)(74,79)(75,83)(77,80)(84,85)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 205486);