#72258562

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×AGL(1,ℤ/16) ⋊ C₂
Sylow	C₃ ⋊ #496124

Subgroups

	subgroup	G/H
center	C₂²	D₃×#442
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₂×C₃×AGL(1,ℤ/16)	C₂
socle	C₂²×C₃	C₂×#442
prime radical	#10950613	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	C₂×AGL(1,ℤ/16)	D₃
hypocenter	C₃	#496124
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×AGL(1,ℤ/16)

1662 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:

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

Polycyclic presentation:

2^9,3|a^2=f,c^2=e,d^2=g,f^2=hi,h^2=i,b^a=bfi,c^a=ch,d^b=dg,e^a=ei,f^b=fh,f^c=fi,h^b=hi,j^d=j^2

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#496124
maximal normal subgroup of	D₃×#496124

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