#72270639

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×AGL(1,ℤ/16) ⋊ C₂
Sylow	C₃ ⋊ #7623148

Subgroups

	subgroup	G/H
center	C₂	C₂×#79869
commutator, cosocle	C₂×C₃×C₈	C₂⁵
Frattini	C₂×C₈	C₂⁴×D₃
Fitting	C₂×C₃×AGL(1,ℤ/16)	C₂
socle	C₂×C₃	C₂×#3109
hypercenter	C₂×AGL(1,ℤ/16)	D₃
hypocenter	C₃	#7623148
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×AGL(1,ℤ/16)

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

Polycyclic presentation:

2^9,3|b^2=f,d^2=gi,f^2=hi,h^2=i,b^a=bg,c^a=cgi,c^b=cf,d^b=dh,e^a=ei,f^a=fi,f^c=fh,f^d=fi,g^b=gi,h^c=hi,j^a=j^2

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#7623148, #72271782
maximal normal subgroup of	D₃×#7623148, #72271782

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