#72225877

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×#3184 ⋊ C₂
Sylow	C₃ ⋊ #10576942

Subgroups

	subgroup	G/H
center	C₂²	C₂³×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁶
Frattini	C₂×C₄	C₂⁵×D₃
Fitting	C₂×C₃×#3184	C₂
socle	C₂²×C₃	C₂⁴×D₄
hypercenter	C₂×#3184	D₃
hypocenter	C₃	#10576942
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#3184

13600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 6 generators on 27 points:

(1,2,3)(13,18)(17,23)(19,25)(24,27);(12,14,15,21,16,20,22,26)(13,17,19,24,18,23,25,27);(2,3)(5,10)(9,11)(12,13)(14,17)(15,19)(16,18)(20,23)(21,24)(22,25)(26,27);(4,7,8,6)(5,9,10,11)(14,20)(17,23)(21,26)(24,27);(4,6,8,7)(5,9,10,11)(14,21)(15,22)(17,24)(19,25)(20,26)(23,27);(4,5)(6,11)(7,9)(8,10)(14,20)(17,23)(21,26)(24,27)

Polycyclic presentation:

2^9,3|b^2=g,c^2=g,e^2=h,h^2=i,b^a=bg,d^a=dg,e^a=ei,e^b=eh,e^c=ei,f^d=fi,h^b=hi,j^d=j^2

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

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