#72254140

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#10742 ⋊ C₂
Sylow	C₃ ⋊ #147817

Subgroups

	subgroup	G/H
center	C₂²	#69598
commutator	C₂²×C₃×D₄	C₂²×C₄
Frattini	C₂²×#41	C₂²×D₃
Fitting	C₃×#10742	C₂
socle	C₂²×C₃	#1790
prime radical	#11016711	C₂
cosocle	C₂²×C₃×#41	C₂³
hypercenter	#10742	D₃
hypocenter	C₃	#147817
LCS	G ▷ C₂²×C₃×D₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #10742

2412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72254153, #72254274, D₃×#147817
maximal normal subgroup of	#72254153, #72254274, D₃×#147817

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