#72270232

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#60698 ⋊ C₂
Sylow	C₃ ⋊ #7623148

Subgroups

	subgroup	G/H
center	C₂	C₂×#83300
commutator, cosocle	C₂×C₃×C₈	C₂⁵
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₃×#60698	C₂
socle	C₂×C₃	C₂×#3109
hypercenter	#60698	D₃
hypocenter	C₃	#7623148
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ #60698

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

Polycyclic presentation:

2^9,3|d^2=g,f^2=hi,h^2=i,c^a=cg,c^b=cf,d^b=dh,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^b=j^2

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72270277, D₃×#7623148
maximal normal subgroup of	#72270277, 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);;
gap> SetPower(coll, 6, F.8*F.9);;
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);;
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, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);