#72242009

order	5616
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₁₃⋊C₃ ⋊ C₂²⋊C₄
Sylow	C₁₃ ⋊ C₃×(C₂×D₃)⋊Dic₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₃⋊AGL(1,𝔽₁₃)
commutator, Fitting, socle	C₂×C₃²×C₁₃	C₂×C₃×C₄
prime radical	C₂×C₃×D₃×D₁₃⋊C₃	C₂
cosocle	C₂×C₃²×D₁₃	C₂²×C₃
hypocenter	C₃²×C₁₃	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃²×C₁₃ ▷ C₃²×C₁₃
UCS	C₁ ◁ C₂

420 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 156

Polycyclic presentation:

2^2,3,2^2,3^2,13|a^2=d,b^2=de,b^a=be,f^a=f^2,f^b=f^2,g^a=f^2g,g^b=g^2,h^a=h^8,h^b=h^8,h^c=h^3,h^d=h^12

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242011, #72242015, #626×AGL(1,𝔽₁₃)
maximal normal subgroup of	#72242011, #72242015, #626×AGL(1,𝔽₁₃)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 3, 3, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^8);;
gap> SetConjugate(coll, 8, 2, F.8^8);;
gap> SetConjugate(coll, 8, 3, F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);