#72271704

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13416 ⋊ C₂
Sylow	C₃ ⋊ #499261

Subgroups

	subgroup	G/H
center	C₄	#81560
commutator, cosocle	C₂²×C₃×C₈	C₂⁴
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13416	C₂
socle	C₂×C₃	#23573
hypercenter	#13416	D₃
hypocenter	C₃	#499261
LCS	G ▷ C₂²×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #13416

1674 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)(5,12,14,23)(6,16,18,24)(7,20)(13,30,31,28)(15,32)(17,34,35,19)(21,22)(29,33);(1,2,3)(5,12,14,23)(6,16,18,24)(7,20)(11,25)(13,30,31,28)(17,34,35,19)(21,22)(26,27);(4,6,8,18)(5,11)(7,19,20,34)(9,16,10,24)(12,26)(13,29)(14,25)(15,28)(17,21,35,22)(23,27)(30,32)(31,33);(4,5,7,13,10,12,22,28,8,14,20,31,9,23,21,30)(6,15,17,27,16,29,19,25,18,32,35,26,24,33,34,11)

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271714
maximal normal subgroup of	#72271714

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