#72271724

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#33638 ⋊ C₂
Sylow	C₃ ⋊ #500473

Subgroups

	subgroup	G/H
center	C₂	#10829230
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#33638	C₂
socle	C₂×C₃	#24128
hypercenter	#33638	D₃
hypocenter	C₃	#500473
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #33638

1534 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)(6,17,19,16)(7,21)(8,9)(11,25,27,35)(12,28)(13,24)(15,31,26,32)(18,33,20,34)(22,23)(29,30);(1,2,3)(6,16,19,17)(7,21)(8,9)(11,25,27,35)(12,28)(13,24)(15,31,26,32)(18,34,20,33)(22,23)(29,30);(4,6,7,16,9,20,21,34,10,19,23,17,8,18,22,33)(5,11,12,25,13,26,28,31,14,27,29,35,24,15,30,32);(4,5)(6,15,19,26)(7,12)(8,24)(9,13)(10,14)(11,18,27,20)(16,32,17,31)(21,28)(22,30)(23,29)(25,33,35,34)

Polycyclic presentation:

2^9,3|a^2=g,b^2=e,c^2=g,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bgh,c^a=cg,c^b=cehi,d^b=defi,e^c=eh,e^d=eh,f^a=fi,f^c=fi,f^d=fi,g^b=gi,h^c=hi,h^d=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.7);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
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, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);