#72259180

order	2496
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₁₃×#99 ⋊ C₂×C₃
Hall	#11718402 ⋊ C₃
Hall	C₁₃⋊C₃ ⋊ B(4,𝔽₂)
Sylow	C₁₃ ⋊ C₃×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#88136⋊C₃
commutator, cosocle	C₂³×C₁₃	C₂³×C₃
Frattini	C₂³	C₂²×D₁₃⋊C₃
Fitting	C₁₃×#99	C₂×C₃
socle	C₂×C₁₃	C₃×#120
hypercenter	#99	D₁₃⋊C₃
hypocenter	C₁₃	C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #99

396 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^3,3,2^3,13|a^2=g,b^a=bf,c^b=ce,e^a=eg,f^c=fg,h^b=h^12,h^c=h^12,h^d=h^3

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	B(4,𝔽₂)×D₁₃⋊C₃
maximal normal subgroup of	B(4,𝔽₂)×D₁₃⋊C₃

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 13]);;
gap> SetPower(coll, 1, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.8^12);;
gap> SetConjugate(coll, 8, 3, F.8^12);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);