#72228219

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×#442 ⋊ C₂²
Sylow	C₃² ⋊ C₂×#60176

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₃²×D₄
commutator	C₃²×C₄	C₂⁵×C₄
Frattini	C₂×C₄	C₂⁴×D₃²
Fitting	C₂×C₃²×#442	C₂²
socle	C₂²×C₃²	C₂⁴×D₄
prime radical	C₂×#11918798	C₂
cosocle	C₂×C₃²×C₄	C₂⁶
hypercenter	C₂×#442	D₃²
hypocenter	C₃²	C₂×#60176
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#442

At least 6104 conjugacy classes of subgroups; only the 7540 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 6 generators on 26 points:

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

Polycyclic presentation:

2^9,3^2|b^2=g,c^2=g,f^2=h,h^2=i,d^a=di,f^b=fh,f^c=fi,h^b=hi,j^a=j^2,j^e=j^2,k^a=j^2k,k^e=k^2

Conjugacy classes & character table

372 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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 5, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 5, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);