#72220563

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃⁴ ⋊ C₂×C₈
Sylow	C₃⁴ ⋊ C₂×C₂²⋊C₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#13049788
commutator, socle	C₂×C₃⁴	C₂²×C₈
Fitting	C₂²×C₃⁴	C₂×C₈
prime radical	C₂²×C₃⋊D₃²	C₄
cosocle	C₂×#66472	C₂³
hypercenter	C₂²	#13049788
hypocenter	C₃⁴	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₂²

1414 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 1414 conjugacy classes of subgroups — too many to draw. Only its 76 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:

2^6,3^4|a^2=d,d^2=f,b^a=be,g^a=h,g^c=g^2,g^d=gh^2,g^f=g^2,h^a=gh^2,h^c=h^2,h^d=g^2h^2,h^f=h^2,i^a=h^2j,i^c=g^2h^2i,i^d=hij^2,i^f=i^2,j^a=gij^2,j^c=g^2hj,j^d=g^2i^2j^2,j^f=j^2

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220584, #72220834, #72220850
maximal normal subgroup of	#72220584, #72220834, #72220850

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