#72247965

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#12756 ⋊ C₂
Sylow	C₃ ⋊ #511827

Subgroups

	subgroup	G/H
center	C₂	#10943959
commutator, cosocle	C₂×C₃×#41	C₂⁴
Frattini	C₂×#41	C₂³×D₃
Fitting	C₃×#12756	C₂
socle	C₂×C₃	#32898
hypercenter	#12756	D₃
hypocenter	C₃	#511827
LCS	G ▷ C₂×C₃×#41 ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #12756

4830 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:

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

Polycyclic presentation:

2^9,3|a^2=e,b^2=efh,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=cgh,d^a=defghi,d^b=defghi,d^c=dh,e^b=ei,e^c=ei,e^d=ei,f^c=fh,f^d=fh,g^b=ghi,g^d=ghi,g^f=gi,h^a=hi,h^b=hi,j^b=j^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#511827
maximal normal subgroup of	D₃×#511827
maximal quotient of	#72270026

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