#72242011

order	11232
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Hall	C₃²×C₁₃⋊C₃ ⋊ C₂×C₂²⋊C₄
Sylow	C₁₃ ⋊ C₃×#63921

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²×AGL(1,𝔽₁₃)
commutator, Fitting, socle	C₂×C₃²×C₁₃	C₂²×C₃×C₄
prime radical	C₂×D₃²×D₁₃⋊C₃	C₂
cosocle	C₂×C₃²×D₁₃	C₂³×C₃
hypocenter	C₃²×C₁₃	C₂×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃²×C₁₃ ▷ C₃²×C₁₃
UCS	C₁ ◁ C₂

1332 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1332 conjugacy classes of subgroups — too many to draw. Only its 148 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:

2^3,3,2^2,3^2,13|a^2=e,b^2=ef,b^a=bf,g^a=g^2,g^c=g^2,h^a=g^2h,h^c=h^2,i^a=i^8,i^b=i^8,i^d=i^3,i^e=i^12

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#3811×AGL(1,𝔽₁₃)
maximal normal subgroup of	#3811×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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 3, 3, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^8);;
gap> SetConjugate(coll, 9, 2, F.9^8);;
gap> SetConjugate(coll, 9, 4, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);