#72259356

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×3₋¹⁺⁴ ⋊ C₂²
Sylow	3₋¹⁺⁴ ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×(#4207⋊C₂)⋊C₃
commutator, cosocle	C₂×#737	C₂³×C₃
Frattini, socle	C₂×C₃	C₂×C₃×#1113
Fitting	C₂²×3₋¹⁺⁴	C₂²
hypercenter	C₂²	(#4207⋊C₂)⋊C₃
hypocenter	#737	C₂×C₃×D₄
LCS	G ▷ C₂×#737 ▷ #737
UCS	C₁ ◁ C₂ ◁ C₂²

1137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 108

Polycyclic presentation:

2^3,3,2,3^4|a^2=e,g^3=i,b^a=be,d^a=di^2,f^a=f^2,f^c=f^2,g^a=g^2i^2,g^d=gi^2,h^c=h^2,h^f=hi^2,i^a=i^2

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×(#4207⋊C₂)⋊C₃, #72276060
maximal normal subgroup of	D₄×(#4207⋊C₂)⋊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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);