#72260344

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#385 ⋊ C₂²
Sylow	C₃² ⋊ C₄×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂²	#10595278
commutator	C₂³×C₃²	C₂³×C₄
Frattini	C₂⁴	C₂²×D₃²
Fitting	C₃²×#385	C₂²
socle	C₂²×C₃²	C₂×#120
prime radical	C₂×#10590410	C₂
cosocle	C₂⁴×C₃²	C₂⁴
hypercenter	#385	D₃²
hypocenter	C₃²	C₄×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #385

4432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 18 points:

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

Polycyclic presentation:

2^8,3^2|a^2=g,b^2=fgh,c^2=e,d^2=e,b^a=bf,c^a=cgh,c^b=cfgh,f^a=fh,f^b=fh,g^b=gh,i^a=i^2,i^d=i^2,j^a=i^2j,j^d=j^2

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.6*F.7*F.8);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 4, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);