#72241999

order	12168
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Hall	C₁₃⋊C₃² ⋊ D₄
Hall	#13062579 ⋊ C₃²
Sylow	C₁₃² ⋊ C₃²×D₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃⋊C₃²
commutator, Fitting, socle, cosocle	C₂×C₁₃²	C₂²×C₃²
hypocenter	C₁₃²	C₃²×D₄
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂

212 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 50 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:

2^2,3^2,2,13^2|b^a=be,f^a=f^12,f^b=f^12,f^c=f^3,f^d=f^9,g^a=f^7g,g^b=g^12,g^c=f^6g,g^d=g^9

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242022
maximal normal subgroup of	#72242022

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");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 2, 13, 13]);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 6, 2, F.6^12);;
gap> SetConjugate(coll, 6, 3, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^9);;
gap> SetConjugate(coll, 7, 1, F.6^7*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^12);;
gap> SetConjugate(coll, 7, 3, F.6^6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);