#72220584

order	10368
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ C₄×C₈
Sylow	C₃⁴ ⋊ C₈×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#72213199
commutator, socle	C₂×C₃⁴	C₂×C₄×C₈
Fitting	C₂²×C₃⁴	C₄×C₈
prime radical	C₂²×C₃⋊D₃²	C₂×C₄
cosocle	C₂×#10620198	C₂³
hypercenter	C₂²	#72213199
hypocenter	C₃⁴	C₈×C₂²⋊C₄
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

1452 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 1452 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,3^4|a^2=f,c^2=d,d^2=g,b^a=be,h^a=hi,h^c=i^2,h^d=hi,h^f=h^2,h^g=h^2,i^a=hi^2,i^c=h^2i^2,i^d=hi^2,i^f=i^2,i^g=i^2,j^a=h^2j,j^c=k,j^d=jk^2,j^f=hi^2j,j^g=j^2,k^a=ik,k^c=jk^2,k^d=j^2k^2,k^f=hk,k^g=k^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222932
maximal normal subgroup of	#72222932

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.9^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8^2);;
gap> SetConjugate(coll, 8, 7, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 3, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 4, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 9, 7, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.8^2*F.10);;
gap> SetConjugate(coll, 10, 3, F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11^2);;
gap> SetConjugate(coll, 10, 6, F.8*F.9^2*F.10);;
gap> SetConjugate(coll, 10, 7, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.9*F.11);;
gap> SetConjugate(coll, 11, 3, F.10*F.11^2);;
gap> SetConjugate(coll, 11, 4, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 6, F.8*F.11);;
gap> SetConjugate(coll, 11, 7, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);