#72241987

order	5616
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃²×C₁₃ ⋊ C₂²×C₃
Hall	C₃²×C₁₃⋊C₃ ⋊ C₂×D₄
Sylow	C₁₃ ⋊ C₃×#3811

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×C₃⋊(D₁₃⋊C₃)
commutator, socle, cosocle	C₂×C₃²×C₁₃	C₂³×C₃
Fitting	C₂²×C₃²×C₁₃	C₂²×C₃
hypercenter	C₂²	D₃×C₃⋊(D₁₃⋊C₃)
hypocenter	C₃²×C₁₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₃²×C₁₃ ▷ C₃²×C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²

580 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₃⋊C₃×#3811
maximal normal subgroup of	D₁₃⋊C₃×#3811

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, 2, 3, 2, 3, 3, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 3, F.8^12);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);