#72227931

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Splittings

Fitting, socle

C₃⁴ ⋊ #226

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, prime radical, cosocle, hypocenter	#14519481	C₂
Fitting, socle	C₃⁴	#226
LCS	G ▷ #14519481
UCS	C₁

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 18

A minimal faithful permutation representation — 2 generators on 18 points:

(1,9,15,18,4,17,3,2)(5,12,10,8,11,6,13,7)(14,16);(3,4,13)(5,14,11)(6,7,9)(8,17,12)

Polycyclic presentation:

2,3,2^3,3^4|a^2=e,c^2=e,d^2=e,b^a=b^2,c^a=cd,c^b=d,d^a=de,d^b=cde,d^c=de,f^a=fh^2i,f^b=g,f^c=g,f^d=fg^2,f^e=f^2,g^a=g^2h,g^b=f^2g^2,g^c=f^2,g^d=f^2g^2,g^e=g^2,h^a=gh,h^b=g^2h^2i,h^c=fg^2h^2i^2,h^d=gi,h^e=h^2,i^a=fgh^2i^2,i^b=f^2g^2h^2,i^c=g^2h^2i,i^d=fgh^2,i^e=i^2

Conjugacy classes & character table

15 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227932

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