#27279408

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #9068

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133186
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2986	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂³×C₅₃	#123
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2986	D₅₃
hypocenter	C₅₃	#9068
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2986

404 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 107 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)(54,56,58,63)(55,60,61,69)(57,65,66,62)(59,67,64,68)(70,73)(71,75)(72,79)(74,77)(76,81)(78,82)(80,83)(84,85);(54,55,57,64,58,61,66,59)(56,60,65,68,63,69,62,67)(70,72,74,80,75,81,82,85)(71,76,78,84,73,79,77,83);(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)(55,59,61,64)(56,62,63,65)(70,71,74,78,75,73,82,77)(72,76,80,84,81,79,85,83)

Polycyclic presentation:

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

Conjugacy classes & character table

1352 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 64884);