#72254198

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#2134 ⋊ C₂
Sylow	C₃ ⋊ #494622

Subgroups

	subgroup	G/H
center	C₂²	C₂×#4741
commutator	C₂×C₃×Q₈	C₂³×C₄
Frattini	C₂²×Q₈	C₂³×D₃
Fitting	C₂×C₃×#2134	C₂
socle	C₂²×C₃	C₂×B(4,𝔽₂)
prime radical	C₂×#83395	C₂
cosocle	C₂²×C₃×Q₈	C₂⁴
hypercenter	C₂×#2134	D₃
hypocenter	C₃	#494622
LCS	G ▷ C₂×C₃×Q₈ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#2134

4056 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)(6,13)(8,15)(9,10)(11,20)(12,14)(16,19)(17,18)(21,22,23);(1,3,4,2)(6,12,14,13)(8,17)(11,19,16,20)(15,18)(22,23);(1,2,4,3)(5,6,9,13,10,14,7,12)(8,16,15,20,17,11,18,19);(6,11,14,16)(7,9)(12,20,13,19)(15,18)

Polycyclic presentation:

2^9,3|a^2=hi,b^2=g,c^2=eg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=chi,c^b=ce,d^a=di,d^b=dhi,d^c=dhi,e^a=eh,e^b=ei,e^d=ei,f^b=fhi,f^c=fh,f^e=fi,g^a=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	#72254214, D₃×#494622, #72254320
maximal normal subgroup of	#72254214, D₃×#494622, #72254320

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.7);;
gap> SetPower(coll, 3, F.5*F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, 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.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
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, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, 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);