#72248328

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#3019 ⋊ C₂
Sylow	C₃ ⋊ #498927

Subgroups

	subgroup	G/H
center	C₂²	C₂×#5032
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₂×C₃×#3019	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	C₂×#3019	D₃
hypocenter	C₃	#498927
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂×#3019

5570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248347, #72248475, D₃×#498927
maximal normal subgroup of	#72248347, #72248475, D₃×#498927
maximal quotient of	#72270439

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