#72277482

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14546455 ⋊ C₂
derived	C₂₄₃ ⋊ C₂×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₂₄₃	C₂×C₉
Frattini	C₃×C₈₁	C₃×D₃
Fitting	#14546455	C₂
socle	C₃	#10624499⋊C₂
prime radical	D₂₄₃⋊C₃	C₃
cosocle	C₂₄₃⋊C₃	C₂×C₃
LCS	G ▷ C₂₄₃
UCS	C₁

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 243

Polycyclic presentation:

2,3^7|b^3=c,d^3=e,e^3=f,f^3=g,g^3=h,d^a=d^2e^2f^2g^2h^2,d^b=dg^2h,d^c=dh^2,e^a=e^2f^2g^2h^2,e^b=eh^2,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277484
maximal normal subgroup of	#72277484

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