Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C3×#398 ⋊ C2 |
| Sylow | C3 ⋊ #2593 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#398 | C2 |
| prime radical | C22×D12 | C22 |
| hypercenter | #398 | D3 |
| hypocenter | C3 | #2593 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #398 | |
440 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(2,3)(5,9)(8,10)(13,16)(17,19)(20,22,23,26)(21,25,24,27);(1,2,3)(12,14,15,18)(13,17,16,19)(21,24)(25,27);(4,6,7,11)(5,8,9,10)(13,16)(14,18);(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,16)(18,19)(20,21)(22,25)(23,24)(26,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=g,d^2=f,b^a=beg,c^a=cf,d^a=deg,c^b=cg,h^d=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707860, #10707864, #10708344, … |
| maximal normal subgroup of | #10707860, #10707864, #10708344, … |
| maximal quotient of | #10717615, #10718720, #10718810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10384);