Finite Group DatabasebetaFinite Group Databasebeta| order | 2888 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C192 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D192 |
| commutator, hypocenter | C192 | C2×C4 |
| Fitting, socle, cosocle | C2×C192 | C22 |
| prime radical | C2×C19⋊D19 | C2 |
| LCS | G ▷ C192 | |
| UCS | C1 ◁ C2 | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 76 |
A minimal faithful permutation representation — 2 generators on 42 points:
(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);(1,2,3,4)(5,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)Polycyclic presentation:
2^3,19^2|a^2=c,d^a=d^18,d^b=d^18,e^a=d^2e,e^b=e^18242 conjugacy classes — view the full character table.
| maximal subgroup of | #72227341, #72227354 |
| maximal normal subgroup of | #72227341, #72227354 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 19, 19]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 4, 2, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);