#72241925

order	3744
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₁₃ ⋊ C₂×C₃×C₄
Hall	C₃×C₁₃⋊C₃ ⋊ C₄×D₄
Sylow	C₁₃ ⋊ C₃×#893

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×AGL(1,𝔽₁₃)
commutator, socle	C₂×C₃×C₁₃	C₂²×C₃×C₄
Fitting	C₃×C₄×C₁₃	C₂×C₃×C₄
prime radical	D₁₂×D₁₃⋊C₃	C₂
cosocle	C₂×C₃×D₁₃	C₂³×C₃
hypercenter	C₄	D₃×AGL(1,𝔽₁₃)
hypocenter	C₃×C₁₃	C₃×C₄×D₄
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

402 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 156

Polycyclic presentation:

2^3,3,2^2,3,13|a^2=f,c^2=e,b^a=bf,c^a=cf,g^b=g^2,h^c=h^8,h^d=h^3,h^e=h^12

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₄×AGL(1,𝔽₁₃), #72242017
maximal normal subgroup of	D₃×D₄×AGL(1,𝔽₁₃)
maximal quotient of	#72242017

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