Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2983 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#431 | C2 |
| prime radical | C23×C3 | C22×C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #431 | D3 |
| hypocenter | C3 | #2983 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #431 | |
356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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:
(1,2,3)(13,19)(17,20)(18,27)(25,26);(2,3)(5,8)(9,11)(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26);(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25);(4,5,6,9,7,8,10,11)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=e,f^2=g,b^a=be,c^a=cg,d^a=de,c^b=ce,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10689062, #10689065, #10689854, … |
| maximal normal subgroup of | #10689062, #10689065, #10689854, … |
| maximal quotient of | #10793920, #10793934, #10794229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11840);