#27287149

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10356

Subgroups

	subgroup	G/H
center	C₂³	#13133187
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1296	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#1296	D₅₃
hypocenter	C₅₃	#10356
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 85 points:

(1,2)(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)(63,66)(64,69)(67,68)(70,72,73,82)(71,77,76,84)(74,81,75,83)(78,85,79,80);(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)(54,56,57,61)(55,58,59,60)(71,76)(72,81)(74,75)(77,80)(82,83)(84,85);(54,55,57,59)(56,60,61,58)(62,63)(64,66)(65,68)(67,69)(70,71,73,76)(72,80,82,85)(74,79,75,78)(77,81,84,83)

Polycyclic presentation:

2^8,53|a^2=d,b^2=dh,c^2=h,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fh,i^b=i^52,e^c=eg,f^c=fh,i^c=i^52

Conjugacy classes & character table

1034 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 72625);