#72249172

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#737 ⋊ C₂×C₃
derived	C₃×C₇×C₉ ⋊ C₂×C₃²
Hall	#13133670 ⋊ C₂
Sylow	C₇ ⋊ #4207⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₇×C₉	C₂×C₃²
Frattini	C₃	C₃×C₃⋊D₃×ΓL(1,𝔽₈)
Fitting	C₇×#737	C₂×C₃
socle	C₃×C₇	C₃²×C₃⋊D₃
LCS	G ▷ C₃×C₇×C₉
UCS	C₁

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

normal subgroups only

This group has 362 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) = 189

Polycyclic presentation:

2,3^4,7,3|e^3=g,c^a=cg^2,d^a=d^2,d^c=dg^2,e^a=e^2g^2,e^b=eg,e^c=eg^2,f^b=f^2,g^a=g^2

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#4207⋊C₃, #72269673
maximal normal subgroup of	ΓL(1,𝔽₈)×#4207⋊C₃, #72269673

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