#27411972

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #23530

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133189
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#575	C₂
socle	C₂²×C₅₃	C₂×#121
hypercenter	C₂×#575	D₅₃
hypocenter	C₅₃	#23530
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#575

1350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(58,62,64,63)(59,67,69,68)(60,72,74,73)(61,77,79,78)(65,70,82,81)(66,75,84,83)(71,86,76,87)(80,85,88,89);(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)(55,57)(58,61,62,77,64,79,63,78)(59,66,68,83,69,84,67,75)(60,71,73,87,74,76,72,86)(65,80,70,85,82,88,81,89);(58,60)(59,65)(61,76)(62,72)(63,73)(64,74)(66,80)(67,70)(68,81)(69,82)(71,79)(75,85)(77,87)(78,86)(83,89)(84,88);(54,55)(56,57)(58,59)(60,70)(61,75)(62,67)(63,68)(64,69)(65,73)(66,78)(71,80)(72,82)(74,81)(76,88)(77,84)(79,83)(85,86)(87,89)

Polycyclic presentation:

2^8,53|c^2=e,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,f^b=fh,g^c=gh,i^c=i^52

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 197448);