#27286703

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10277

Subgroups

	subgroup	G/H
center	C₂³	#13133188
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#487	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	C₂×#487	D₅₃
hypocenter	C₅₃	#10277
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 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)(60,69)(61,62)(64,68)(65,66)(70,71)(72,73)(74,76,78,84)(75,79,81,88)(77,86,87,83)(80,89,85,82);(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)(58,60,63,71)(59,64,67,73)(61,70,62,69)(65,72,66,68);(54,55)(56,57)(58,59,63,67)(60,68,71,72)(61,66,62,65)(64,70,73,69)(74,75,78,81)(76,82,84,89)(77,85,87,80)(79,86,88,83)

Polycyclic presentation:

2^8,53|a^2=gh,b^2=g,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,f^b=fg,i^b=i^52,d^c=dg,e^c=eh,f^c=fg

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, 72179);