#72249073

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Hall	#13133487 ⋊ C₂
Hall	D₇ ⋊ #6875
Sylow	C₇ ⋊ C₂×#6875

Subgroups

	subgroup	G/H
center	C₃	(D₇⋊C₉)⋊C₉
commutator	C₇×C₉	C₂×C₃×C₉
Frattini	C₃×C₉	C₃×AGL(1,𝔽₇)
Fitting	C₇×C₂₇⋊C₃	C₂×C₃
socle	C₃×C₇	C₂×C₉⋊C₉
prime radical	C₃²×D₇	3₋¹⁺²
cosocle	C₃×C₇×C₉	C₂×C₃²
hypercenter	C₂₇⋊C₃	AGL(1,𝔽₇)
hypocenter	C₇	C₂×#6875
LCS	G ▷ C₇×C₉ ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₂₇⋊C₃

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 189

Polycyclic presentation:

2,3^5,7|b^3=d,c^3=e^2f,d^3=f,e^3=f^2,c^b=ce,d^c=df,e^b=ef,g^a=g^6,g^c=g^2

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#6875
maximal normal subgroup of	AGL(1,𝔽₇)×#6875

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