#72259595

order	32448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	#72259288 ⋊ C₃
Hall	#92788 ⋊ #417
Sylow	C₁₃² ⋊ C₃×#417

Subgroups

	subgroup	G/H
center	C₂	C₂×#72214186
commutator	C₄×C₁₃²	C₂²×C₃×C₄
Frattini	C₄	C₂×#72210302
Fitting	C₂×C₄×C₁₃²	C₂×C₃×C₄
socle	C₂×C₁₃²	C₂×C₃×C₂²⋊C₄
prime radical	#72259381	C₂
cosocle	C₄×C₁₃⋊D₁₃	C₂³×C₃
hypercenter	C₂×C₄	#72210302
hypocenter	C₁₃²	C₃×#417
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

1200 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 106 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:

2^3,3,2^3,13^2|a^2=e,b^2=ef,e^2=g,f^2=g,b^a=bf,c^a=cg,e^b=eg,f^b=fg,h^a=h^5,h^b=h^8,h^d=h^3,h^e=h^12,i^a=h^6i^8,i^b=i^8,i^d=i^3,i^e=i^12

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259662
maximal normal subgroup of	#72259662

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^5);;
gap> SetConjugate(coll, 8, 2, F.8^8);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> SetConjugate(coll, 8, 5, F.8^12);;
gap> SetConjugate(coll, 9, 1, F.8^6*F.9^8);;
gap> SetConjugate(coll, 9, 2, F.9^8);;
gap> SetConjugate(coll, 9, 4, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);