#72247101

order	7182
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₁₉ ⋊ C₂×C₃×C₉
Hall	ΓL(1,𝔽₈)×#4296 ⋊ C₂
Hall	D₁₃₃ ⋊ C₃×C₉
Sylow	C₁₉ ⋊ C₉×AGL(1,𝔽₇)
Sylow	C₇ ⋊ C₃×AGL(1,𝔽₁₉)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₁₉	C₂×C₃×C₉
prime radical	ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	C₃
cosocle	C₇×C₁₉⋊C₃	C₂×C₃²
LCS	G ▷ C₇×C₁₉
UCS	C₁

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 133

Polycyclic presentation:

2,3^3,7,19|c^3=d,e^a=e^6,e^b=e^4,f^a=f^18,f^c=f^9,f^d=f^7

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×AGL(1,𝔽₁₉)
maximal normal subgroup of	AGL(1,𝔽₇)×AGL(1,𝔽₁₉)

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, 3, 3, 3, 7, 19]);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 6, 3, F.6^9);;
gap> SetConjugate(coll, 6, 4, F.6^7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);