#72247883

order	4048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₁ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₈×C₂₃ ⋊ C₂×C₁₁
Hall	D₁₈₄ ⋊ C₁₁
Hall	C₂₃⋊C₁₁ ⋊ D₈
Sylow	C₂₃ ⋊ C₁₁×D₈

Subgroups

	subgroup	G/H
center	C₂	#72222400
commutator, cosocle	C₄×C₂₃	C₂²×C₁₁
Frattini	C₄	C₂×AGL(1,𝔽₂₃)
Fitting	C₈×C₂₃	C₂×C₁₁
socle	C₂×C₂₃	D₄×C₁₁
hypercenter	C₈	AGL(1,𝔽₂₃)
hypocenter	C₂₃	C₁₁×D₈
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 184

Polycyclic presentation:

2^2,11,2^2,23|a^2=d,d^2=e,b^a=bd,d^b=de,f^b=f^22,f^c=f^18

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247885, #72247901, D₈×AGL(1,𝔽₂₃)
maximal normal subgroup of	#72247885, #72247901, D₈×AGL(1,𝔽₂₃)

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");;
gap> coll := SingleCollector(F, [2, 2, 11, 2, 2, 23]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^22);;
gap> SetConjugate(coll, 6, 3, F.6^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);