Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#2822 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
380 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(6,7)(10,11)(13,14,17,20)(15,18,22,26)(16,23,24,19)(21,27,28,25)(29,30,31);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,15)(14,18)(16,21)(17,22)(19,25)(20,26)(23,27)(24,28);(2,4)(5,7,9,6)(8,10,12,11)(13,16,17,24)(14,19,20,23)(15,21,22,28)(18,25,26,27);(5,8,9,12)(6,10,7,11)(15,22)(18,26)(21,28)(25,27)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17262);