#72258593

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33329 ⋊ C₂
Sylow	C₃ ⋊ #7622637

Subgroups

	subgroup	G/H
center	C₂	C₂²×#4711
commutator	C₃×C₈	C₂⁴×C₄
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#33329	C₂
socle	C₂×C₃	C₂²×#442
prime radical	#11701727	C₂
cosocle	C₂×C₃×C₈	C₂⁵
hypercenter	#33329	D₃
hypocenter	C₃	#7622637
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #2875 ◁ #33329

3826 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)(5,13)(6,14)(11,24)(12,26)(15,27)(16,28)(23,25)(29,35);(5,12,13,26)(7,18)(10,21,22,34)(11,23,24,25)(15,27)(17,30)(20,31,32,33)(29,35);(2,3)(5,11,13,24)(7,17)(8,19)(10,20,22,32)(12,23,26,25)(15,29)(16,28)(18,30)(21,31,34,33)(27,35);(4,6)(5,10)(7,15)(8,16)(9,14)(11,20)(12,21)(13,22)(17,29)(18,27)(19,28)(23,31)(24,32)(25,33)(26,34)(30,35);(4,5,7,11,8,12,17,25,9,13,18,24,19,26,30,23)(6,10,15,20,16,21,29,33,14,22,27,32,28,34,35,31)

Polycyclic presentation:

2^9,3|a^2=g,c^2=f,d^2=f,g^2=h,h^2=i,c^a=cghi,d^a=dhi,e^a=ei,e^b=ei,f^a=fi,g^c=gh,g^d=gi,h^c=hi,j^c=j^2

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7622637, #72270277
maximal normal subgroup of	D₃×#7622637, #72270277

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