#27492402

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1909 ⋊ C₂
Sylow	C₅₃ ⋊ #29903

Subgroups

	subgroup	G/H
center	C₂	#20280492
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1909	C₂
socle	C₂×C₅₃	#2622
prime radical	C₂×#13133263	C₂²
hypercenter	#1909	D₅₃
hypocenter	C₅₃	#29903
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1909

756 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)(54,55)(56,61)(57,62)(58,63)(59,75)(60,64)(65,67)(66,81)(68,77)(69,82)(70,72)(71,78)(73,80)(74,79)(76,84)(83,85);(54,57,60,74)(55,62,64,79)(56,66,67,84)(58,73,59,72)(61,76,65,81)(63,80,75,70)(68,71,69,85)(77,83,82,78);(54,56,58,68,60,67,59,69)(55,61,63,77,64,65,75,82)(57,71,72,66,74,85,73,84)(62,78,70,81,79,83,80,76);(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)(54,55)(56,65)(57,70)(58,75)(59,63)(60,64)(61,67)(62,72)(66,83)(68,77)(69,82)(71,76)(73,79)(74,80)(78,84)(81,85)

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 277878);