#72276149

order	2496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	#11718813 ⋊ C₃
Hall	C₁₃⋊C₃ ⋊ #442
Sylow	C₁₃ ⋊ C₃×#442

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₁₃⋊C₃
commutator	C₄×C₁₃	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×D₁₃⋊C₃
Fitting	C₂×C₁₃×D₈	C₂×C₃
socle	C₂²×C₁₃	C₂×C₃×D₄
prime radical	C₂×D₈×C₁₃⋊C₃	C₂
cosocle	C₂×C₄×C₁₃	C₂³×C₃
hypercenter	C₂×D₈	D₁₃⋊C₃
hypocenter	C₁₃	C₃×#442
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

264 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 416

Polycyclic presentation:

2^3,3,2^3,13|a^2=e,b^2=e,c^2=f,f^2=g,c^a=cf,c^b=cg,f^a=fg,h^a=h^12,h^b=h^12,h^d=h^9

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#442×D₁₃⋊C₃
maximal normal subgroup of	#442×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.5);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> SetConjugate(coll, 8, 2, F.8^12);;
gap> SetConjugate(coll, 8, 4, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);