Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2381 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#391 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | #391 | D3 |
| hypocenter | C3 | #2381 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #391 | |
424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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:
(1,2,3)(9,13)(12,14)(16,19,20,28)(17,23,22,30)(18,25,26,27)(21,29,24,31);(2,3)(8,10,11,15)(9,12,13,14)(17,22)(21,24)(23,30)(29,31);(10,15)(12,14)(16,18,20,26)(17,21,22,24)(19,27,28,25)(23,31,30,29);(4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,17)(18,24)(19,23)(20,22)(21,26)(25,31)(27,29)(28,30)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10712014, #10712017, #10712531, … |
| maximal normal subgroup of | #10712014, #10712017, #10712531, … |
| maximal quotient of | #10716812, #10716834, #10716835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9073);