#72249090

order	6804
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

derived	C₇×C₂₇ ⋊ C₂²×C₉
Hall	#13133485 ⋊ C₂²
Sylow	C₇ ⋊ C₂×AGL(1,ℤ/27)

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₇×C₂₇	C₂²×C₉
Frattini	C₃×C₉	D₃×AGL(1,𝔽₇)
Fitting, cosocle	C₇×C₂₇⋊C₃	C₂²×C₃
socle	C₃×C₇	C₂×D₉⋊C₉
prime radical	D₇×D₂₇⋊C₃	C₃
LCS	G ▷ C₇×C₂₇
UCS	C₁

136 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 36 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:

2^2,3^3,7,3^2|c^3=d,e^3=g,g^3=h,e^a=e^2g^2h^2,e^c=eg^2h^2,e^d=eh^2,f^b=f^6,f^c=f^2,g^a=g^2h^2,g^c=gh^2,h^a=h^2

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×AGL(1,ℤ/27)
maximal normal subgroup of	AGL(1,𝔽₇)×AGL(1,ℤ/27)

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 7, 3, 3]);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 2, F.6^6);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);