#72242010

order	12168
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃×C₁₃² ⋊ C₂×C₃×C₄
Hall	C₃×#92788 ⋊ C₂×C₄
Sylow	C₁₃² ⋊ C₃×C₄×D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₃²	C₂×C₃×C₄
prime radical	#11759817⋊D₃	C₂
cosocle	C₃×C₁₃×D₁₃	C₂²×C₃
LCS	G ▷ C₃×C₁₃²
UCS	C₁

167 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 32 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:

2^2,3,2,3,13^2|a^2=d,e^b=e^2,f^a=f^5,f^b=f^12,f^c=f^3,f^d=f^12,g^a=f^2g^12,g^b=g^12,g^c=g^3,g^d=f^8g

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72214102
maximal normal subgroup of	D₃×#72214102

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