#72270623

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33659 ⋊ C₂
Sylow	C₃ ⋊ #7623148

Subgroups

	subgroup	G/H
center	C₂	C₂×#81310
commutator, cosocle	C₂×C₃×C₈	C₂⁵
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#33659	C₂
socle	C₂×C₃	C₂×#3109
hypercenter	#33659	D₃
hypocenter	C₃	#7623148
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #33659

2976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 35 points:

(1,2,3)(5,11)(10,24)(12,18)(13,26)(22,25)(23,34)(27,31)(33,35);(2,3)(6,15,17,29)(7,20)(10,23,24,34)(12,18)(14,28,16,30)(19,32)(22,33,25,35)(27,31);(6,14,17,16)(7,19)(9,21)(10,22,24,25)(12,27)(13,26)(15,28,29,30)(18,31)(20,32)(23,33,34,35);(4,6,7,16,9,15,19,28,8,17,20,14,21,29,32,30)(5,10,12,25,13,23,27,33,11,24,18,22,26,34,31,35);(4,5)(6,10)(7,18)(8,11)(9,13)(12,20)(14,25)(15,23)(16,22)(17,24)(19,31)(21,26)(27,32)(28,35)(29,34)(30,33)

Polycyclic presentation:

2^9,3|b^2=f,c^2=gi,d^2=gi,f^2=h,h^2=i,b^a=bg,c^a=cgi,c^b=cfh,d^b=dhi,e^a=ei,f^a=fi,f^c=fh,f^d=fi,g^b=gi,h^c=hi,j^d=j^2

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7623148
maximal normal subgroup of	D₃×#7623148

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