#27411950

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23527

Subgroups

	subgroup	G/H
center	C₄	C₂×#13133204
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#575	C₂
socle	C₂×C₅₃	C₂×#394
hypercenter	C₂×#575	D₅₃
hypocenter	C₅₃	#23527
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#575

1336 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)(57,70)(61,68)(65,69)(71,85)(75,81)(77,82)(79,83)(80,84);(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)(54,57,58,71,59,70,74,85)(55,61,62,77,63,68,72,82)(56,65,66,80,67,69,73,84)(60,75,64,79,76,81,78,83);(54,56)(55,60)(57,69)(58,73)(59,67)(61,75)(62,78)(63,76)(64,72)(65,70)(66,74)(68,81)(71,80)(77,83)(79,82)(84,85);(54,55)(56,64)(57,68)(58,72)(59,63)(60,66)(61,70)(62,74)(65,79)(67,78)(69,83)(71,77)(73,76)(75,80)(81,84)(82,85)

Polycyclic presentation:

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

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 197426);