#72258653

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33388 ⋊ C₂
Sylow	C₃ ⋊ #7622637

Subgroups

	subgroup	G/H
center	C₂	C₂²×#4785
commutator	C₃×C₈	C₂⁴×C₄
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#33388	C₂
socle	C₂×C₃	C₂²×#442
prime radical	#11701725	C₂
cosocle	C₂×C₃×C₈	C₂⁵
hypercenter	#33388	D₃
hypocenter	C₃	#7622637
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#131 ◁ #33388

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

Polycyclic presentation:

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7622637, #72271460
maximal normal subgroup of	D₃×#7622637, #72271460

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