Finite Group DatabasebetaFinite Group Databasebeta| order | 19320 |
| automorphism | Aut(G) order 20401920 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D9660 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D2415 |
| commutator, socle, cosocle | C2×C3×C5×C7×C23 | C22 |
| Fitting | C3×C4×C5×C7×C23 | C2 |
| hypercenter | C4 | D2415 |
| hypocenter | C3×C5×C7×C23 | D4 |
| LCS | G ▷ C2×C3×C5×C7×C23 ▷ C3×C5×C7×C23 | |
| UCS | C1 ◁ C2 ◁ C4 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 9660 |
A minimal faithful permutation representation — 2 generators on 42 points:
(1,2,3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42);(2,4)(6,7)(9,12)(10,11)(14,19)(15,18)(16,17)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)Polycyclic presentation:
2^3,3,5,7,23|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^6,g^a=g^224833 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D19320, D28980 |
| maximal normal subgroup of | D19320 |
| maximal quotient of | Dic9660, D19320, D28980 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 7, 23]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);