Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2195 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C3×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#131 ◁ G | |
206 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,8,14)(3,15,19,43)(4,20,23,40)(5,24,28,38)(6,13,7,12)(9,31,34,17)(10,29,36,22)(11,26,39,27)(16,44,48,61)(18,49,30,37)(21,52,55,35)(25,41,56,33)(32,53,60,64)(42,47,63,46)(45,50,58,59)(51,57,62,54)(65,66,67);(1,3)(2,9)(4,16)(5,25)(6,18)(7,30)(8,19)(10,32)(11,37)(12,33)(13,41)(14,34)(15,24)(17,27)(20,50)(21,53)(22,46)(23,48)(26,31)(28,56)(29,47)(35,61)(36,60)(38,43)(39,49)(40,59)(42,51)(44,52)(45,54)(55,64)(57,58)(62,63);(1,4,8,23)(2,10,14,36)(3,16,19,48)(5,21,28,55)(6,29,7,22)(9,32,34,60)(11,35,39,52)(12,40,13,20)(15,42,43,63)(17,45,31,58)(18,47,30,46)(24,51,38,62)(25,53,56,64)(26,57,27,54)(33,59,41,50)(37,61,49,44)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5486);