#72270532

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13079 ⋊ C₂
Sylow	C₃ ⋊ #495573

Subgroups

	subgroup	G/H
center	C₂	#10800419
commutator	C₂×C₃×C₈	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13079	C₂
socle	C₂×C₃	#21408
prime radical	#11665253	C₂
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#13079	D₃
hypocenter	C₃	#495573
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #13079

1564 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:

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

Polycyclic presentation:

2^9,3|b^2=e,d^2=g,e^2=hi,h^2=i,c^a=cg,c^b=cef,d^b=dh,e^c=eh,e^d=ei,f^a=fi,g^b=gi,h^c=hi,j^b=j^2,j^d=j^2

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495573, #72271460
maximal normal subgroup of	D₃×#495573, #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, 2, F.5);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
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.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);