#72249229

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#20579 ⋊ C₂
Sylow	C₃ ⋊ #392113

Subgroups

	subgroup	G/H
center	C₂³	C₂²×(C₂×D₃)⋊C₄
commutator	C₂⁴×C₃	C₂³×C₄
Frattini	C₂³×C₄	C₂³×D₃
Fitting	C₃×#20579	C₂
socle	C₂³×C₃	C₂²×C₂²⋊C₄
prime radical	C₂⁵×D₁₂	C₂
cosocle	C₂³×C₃×C₄	C₂⁴
hypercenter	#20579	D₃
hypocenter	C₃	#392113
LCS	G ▷ C₂⁴×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ #20579

57324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Polycyclic presentation:

2^9,3|b^2=e,d^2=eg,e^2=i,b^a=bh,c^b=cf,d^a=dh,d^b=dg,d^c=df,e^d=ei,g^b=gi,g^d=gi,j^d=j^2

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#392113, #72270477, #72271505, …
maximal normal subgroup of	D₃×#392113, #72270477, #72271505, …

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