#72259490

order	16224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	#72259275 ⋊ C₃
Hall	#92788 ⋊ AGL(1,ℤ/8)
Sylow	C₁₃² ⋊ C₃×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	#72214127
commutator, cosocle	C₄×C₁₃²	C₂³×C₃
Frattini	C₄	C₂×#11759815⋊C₂
Fitting	D₄×C₁₃²	C₂²×C₃
socle	C₂×C₁₃²	C₂×C₃×D₄
hypercenter	D₄	#11759815⋊C₂
hypocenter	C₁₃²	C₃×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

596 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 58 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:

2^3,3,2^2,13^2|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^12,g^c=g^12,g^d=g^3,h^a=g^7h,h^c=h^12,h^d=h^3

Conjugacy classes & character table

121 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259514
maximal normal subgroup of	#72259514

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