#27272912

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #7756

Subgroups

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2023	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂²×D₄×C₅₃	D₄
hypercenter	#2023	D₅₃
hypocenter	C₅₃	#7756
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2023

504 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 6784

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

Polycyclic presentation:

2^8,53|a^2=f,b^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,i^b=i^52,d^c=dg,e^c=eh

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, 58388);