#72249083

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₇×C₂₇⋊C₃ ⋊ C₂×C₃
derived	C₇×C₂₇ ⋊ C₂×C₃²
Hall	ΓL(1,𝔽₈)×C₂₇⋊C₃ ⋊ C₂
Sylow	C₇ ⋊ C₃×D₂₇⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×C₂₇	C₂×C₃²
Frattini	C₉	C₃×C₃⋊AGL(1,𝔽₇)
Fitting	C₇×C₂₇⋊C₃	C₂×C₃
socle	C₃×C₇	C₃²×D₉
LCS	G ▷ C₇×C₂₇
UCS	C₁

132 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 33 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:

2,3^3,7,3^2|d^3=f,f^3=g,d^a=d^2f^2g^2,d^c=dg^2,e^a=e^6,e^b=e^2,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×D₂₇⋊C₃, #72249095
maximal normal subgroup of	AGL(1,𝔽₇)×D₂₇⋊C₃, #72249095

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