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×#2817 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
444 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 444 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) = 192 |
A minimal faithful permutation representation — 4 generators on 31 points:
(5,6)(7,9)(8,10)(11,12)(13,14,18,21)(15,22,17,19)(16,24,26,20)(23,27,25,28)(29,30,31);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,17)(14,19,21,22)(16,25,26,23)(20,28,24,27);(2,4)(6,10)(9,12)(13,16,18,26)(14,20,21,24)(15,23,17,25)(19,27,22,28);(7,11)(9,12)(13,17,18,15)(14,19,21,22)(16,23,26,25)(20,28,24,27)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17257);