#72259376

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#6911 ⋊ C₂²
Sylow	#6911 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×(#4178⋊C₂)⋊C₃
commutator, cosocle	C₂×#732	C₂³×C₃
Frattini	C₂×C₃²	C₂×C₃×D₃²
Fitting	C₄×#6911	C₂²
socle	C₂×C₃	C₂×C₃×((C₃⋊D₃)⋊C₃)⋊C₂
hypercenter	C₄	(#4178⋊C₂)⋊C₃
hypocenter	#732	C₂×C₃×D₄
LCS	G ▷ C₂×#732 ▷ #732
UCS	C₁ ◁ C₂ ◁ C₄

715 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 108

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×(#4178⋊C₂)⋊C₃, #72259991
maximal normal subgroup of	D₄×(#4178⋊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.9^2);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 3, F.7^2*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^2*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9^2);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);