#72242001

order	24336
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃×C₁₃² ⋊ C₃×C₄²
Hall	C₃×#92788 ⋊ C₄²
Sylow	C₁₃² ⋊ C₃×C₄×Dic₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₃²	C₃×C₄²
prime radical	C₃×#11759815⋊C₂	C₂²
cosocle	C₃×D₁₃²	C₂²×C₃
LCS	G ▷ C₃×C₁₃²
UCS	C₁

326 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 58 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:

2^2,3,2^2,3,13^2|a^2=d,b^2=e,f^a=f^2,g^a=g^8,g^b=g^8,g^c=g^3,g^d=g^12,g^e=g^12,h^a=g^6h,h^b=h^8,h^c=h^3,h^d=g^2h,h^e=h^12

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72214122
maximal normal subgroup of	D₃×#72214122

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