#72259128

order	23760
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₃²×C₁₁ ⋊ C₅×GL(2,𝔽₃)
derived	C₁₁×PSU(3,𝔽₂)⋊C₃ ⋊ C₂×C₅
Hall	#72259104 ⋊ C₅
Hall	C₁₁⋊C₅ ⋊ AGL(2,𝔽₃)
Sylow	C₁₁ ⋊ C₅×AGL(2,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₁₁×PSU(3,𝔽₂)⋊C₃	C₂×C₅
Fitting, socle	C₃²×C₁₁	C₅×GL(2,𝔽₃)
LCS	G ▷ C₁₁×PSU(3,𝔽₂)⋊C₃
UCS	C₁

184 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 99

A minimal faithful permutation representation — 2 generators on 20 points:

(1,7,4,6,3,9,5,2)(11,15,16,14,19,12,13,20,17,18);(3,5,6)(7,8,9)(10,11,14,19,13,16,20,15,18,17,12)

Polycyclic presentation:

2,5,3,11,2^3,3^2|e^2=g,f^2=g,c^a=c^2,d^a=d^10,d^b=d^4,e^a=ef,e^c=f,f^a=fg,f^c=efg,f^e=fg,h^a=h^2,h^e=i,h^f=hi,h^g=h^2,i^a=h^2i,i^c=hi,i^e=h^2,i^f=hi^2,i^g=i^2

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₁)×AGL(2,𝔽₃)
maximal normal subgroup of	AGL(1,𝔽₁₁)×AGL(2,𝔽₃)

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, 5, 3, 11, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 4, 2, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.5*F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8^2);;
gap> SetConjugate(coll, 9, 6, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 7, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);