#72277827

order	5928
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₁₃×C₁₉ ⋊ C₂×C₃×C₄
Hall	D₁₉×#261 ⋊ C₃
Hall	#10625203 ⋊ C₂×C₄
Sylow	C₁₉ ⋊ C₂×AGL(1,𝔽₁₃)
Sylow	C₁₃ ⋊ C₄×D₁₉⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₃×C₁₉	C₂×C₃×C₄
prime radical	#13093361⋊C₂	C₂
cosocle	C₁₉×D₁₃	C₂²×C₃
LCS	G ▷ C₁₃×C₁₉
UCS	C₁

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 247

Polycyclic presentation:

2^2,3,2,13,19|a^2=d,e^a=e^5,e^c=e^9,e^d=e^12,f^b=f^18,f^c=f^7

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₁₉⋊C₃×AGL(1,𝔽₁₃)
maximal normal subgroup of	D₁₉⋊C₃×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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 13, 19]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^5);;
gap> SetConjugate(coll, 5, 3, F.5^9);;
gap> SetConjugate(coll, 5, 4, F.5^12);;
gap> SetConjugate(coll, 6, 2, F.6^18);;
gap> SetConjugate(coll, 6, 3, F.6^7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);