Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#578 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C25×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ G | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(3,4,5);(6,7,11,15)(8,16,19,29)(9,13,20,27)(10,22,24,14)(12,23,26,18)(17,30,31,37)(21,34,33,28)(25,35,36,32);(6,8,11,19)(7,12,15,26)(9,17,20,31)(10,23,24,18)(13,25,27,36)(14,29,22,16)(21,35,33,32)(28,37,34,30);(9,20)(13,27)(17,31)(21,33)(25,36)(28,34)(30,37)(32,35);(1,2)(6,9)(7,13)(8,17)(10,21)(11,20)(12,25)(14,28)(15,27)(16,30)(18,32)(19,31)(22,34)(23,35)(24,33)(26,36)(29,37)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20043);