#72271728

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13459 ⋊ C₂
Sylow	C₃ ⋊ #499261

Subgroups

	subgroup	G/H
center	C₄	#81888
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13459	C₂
socle	C₂×C₃	#23573
hypercenter	#13459	D₃
hypocenter	C₃	#499261
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #13459

1674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 35 points:

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=fg,c^2=g,d^2=g,e^2=hi,h^2=i,b^a=bef,c^a=cfh,c^b=cfghi,d^a=dh,d^b=dh,e^b=eh,e^c=ei,e^d=ei,f^b=fi,g^a=gi,g^b=gi,h^b=hi,j^d=j^2

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271743
maximal normal subgroup of	#72271743

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