#27273221

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7802

Subgroups

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2083	C₂
socle, prime radical	C₂³×C₅₃	#120
hypercenter	#2083	D₅₃
hypocenter	C₅₃	#7802
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2083

366 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

869 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 58697);