#72270397

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12605 ⋊ C₂
Sylow	C₃ ⋊ #495505

Subgroups

	subgroup	G/H
center	C₄	#73563
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#12605	C₂
socle	C₂×C₃	#21379
prime radical	#85680	C₄
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#12605	D₃
hypocenter	C₃	#495505
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ #12605

1552 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:

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

Polycyclic presentation:

2^9,3|b^2=e,d^2=g,e^2=h,h^2=i,c^a=cg,c^b=cf,d^b=dhi,e^d=ei,f^a=fi,g^b=gi,j^a=j^2

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#495505
maximal normal subgroup of	D₃×#495505

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