#72270633

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#60184 ⋊ C₂
Sylow	C₃ ⋊ #7623148

Subgroups

	subgroup	G/H
center	C₂	C₂×#82310
commutator, cosocle	C₂×C₃×C₈	C₂⁵
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#60184	C₂
socle	C₂×C₃	C₂×#3109
hypercenter	#60184	D₃
hypocenter	C₃	#7623148
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ #60184

2976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 35 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7623148
maximal normal subgroup of	D₃×#7623148

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