#72270409

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#13200 ⋊ C₂
Sylow	C₃ ⋊ #495505

Subgroups

	subgroup	G/H
center	C₄	#76290
commutator	C₂²×C₃×C₄	C₂³×C₄
Frattini	C₂²×C₈	C₂³×D₃
Fitting	C₃×#13200	C₂
socle	C₂×C₃	#21379
prime radical	#86018	C₄
cosocle	C₂²×C₃×C₈	C₂⁴
hypercenter	#13200	D₃
hypocenter	C₃	#495505
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #13200

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

132 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, 1, F.7);;
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, 2, 1, F.2*F.8);;
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, 1, 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, 2, F.7*F.9);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);