#72214799

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow

C₃⁵ ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃⁵	C₂×C₈
prime radical	C₃×C₃⋊D₃²	C₄
cosocle	C₃×#66472	C₂²
LCS	G ▷ C₃⁵
UCS	C₁

468 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3^5|b^2=c,c^2=d,e^b=e^2,f^a=f^2,f^b=g,f^c=fg^2,f^d=f^2,g^a=g^2,g^b=fg^2,g^c=f^2g^2,g^d=g^2,h^a=g^2h,h^b=i,h^c=hi^2,h^d=h^2,i^a=f^2gi,i^b=hi^2,i^c=h^2i^2,i^d=i^2

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214807, D₃×#13049788, #72224565, …
maximal normal subgroup of	#72214807, D₃×#13049788, #72224565, …

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 3, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9^2);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.6^2*F.7*F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 3, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 4, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);