#72248105

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12583 ⋊ C₂
Sylow	C₃ ⋊ #498888

Subgroups

	subgroup	G/H
center	C₂²	C₂×#4741
commutator, cosocle	C₂²×C₃×D₄	C₂⁴
Frattini	C₂²×D₄	C₂³×D₃
Fitting	C₃×#12583	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
hypercenter	#12583	D₃
hypocenter	C₃	#498888
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #12583

4668 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:

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

Polycyclic presentation:

2^9,3|a^2=f,b^2=e,d^2=i,g^2=i,b^a=bg,c^a=cfghi,c^b=ce,d^a=dhi,d^b=dhi,d^c=di,e^a=ei,f^b=fh,f^c=fh,f^d=fi,g^a=ghi,g^c=gh,g^d=gi,g^f=gi,h^a=hi,j^c=j^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72248142, #72254274, D₃×#498888
maximal normal subgroup of	#72248142, #72254274, D₃×#498888

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