#72270628

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33657 ⋊ 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×#33657	C₂
socle	C₂×C₃	C₂×#3109
hypercenter	#33657	D₃
hypocenter	C₃	#7623148
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #33657

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

Polycyclic presentation:

2^9,3|b^2=f,d^2=gh,f^2=hi,h^2=i,b^a=bg,c^a=cgi,c^b=cfgh,d^a=di,d^b=dh,d^c=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, 4, F.7*F.8);;
gap> SetPower(coll, 6, F.8*F.9);;
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.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, 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);