#72227607

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#104 ⋊ Q₈
Sylow	C₃² ⋊ #25830

Subgroups

	subgroup	G/H
center	C₂	#11812041
commutator, cosocle	C₂×C₄×C₃⋊D₃	C₂⁴
Frattini	C₂×C₄	C₂²×PSU(3,𝔽₂)
Fitting	C₃²×#104	Q₈
socle	C₂×C₃²	#2434
prime radical	#41×C₃⋊D₃	C₂³
hypercenter	#104	PSU(3,𝔽₂)
hypocenter	C₃²	#25830
LCS	G ▷ C₂×C₄×C₃⋊D₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

1139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 4 generators on 17 points:

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

Polycyclic presentation:

2^8,3^2|b^2=g,c^2=g,d^2=g,f^2=h,b^a=bf,c^a=ch,c^b=ce,d^b=dg,d^c=dg,e^a=eh,f^a=fh,f^b=fh,f^c=fh,i^b=i^2j,i^c=i^2j,i^d=j^2,i^g=i^2,j^b=ij,j^c=ij,j^d=i,j^g=j^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#453×PSU(3,𝔽₂)
maximal normal subgroup of	#453×PSU(3,𝔽₂)

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