#72248015

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#3025 ⋊ C₂
Sylow	C₃ ⋊ #498885

Subgroups

	subgroup	G/H
center	C₂×C₄	#5032
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₂×C₃×#3025	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	C₂×#3025	D₃
hypocenter	C₃	#498885
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3025

4652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(5,8,10,20)(6,12,13,9)(7,16,18,19)(11,17,14,15)(21,22,23);(2,4)(7,15)(11,19)(14,16)(17,18);(5,7,12,15)(6,11,8,19)(9,17,10,18)(13,14,20,16);(1,2)(3,4)(5,6)(7,14,18,11)(8,9)(10,13)(12,20)(15,16,17,19)(22,23)

Polycyclic presentation:

2^9,3|a^2=hi,b^2=fgi,d^2=i,g^2=i,b^a=bf,c^a=ce,c^b=cfgi,d^a=di,d^b=dhi,d^c=dhi,f^a=fi,f^b=fh,f^c=fhi,g^a=ghi,g^b=gh,g^c=ghi,g^d=gi,g^f=gi,h^b=hi,j^a=j^2

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248020, D₃×#498885, #72254153, …
maximal normal subgroup of	#72248020, D₃×#498885, #72254153, …
maximal quotient of	#72271616

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