Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C3 ⋊ #2880 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting, prime radical | C22×C3×#41 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | C22×#41 | D3 |
| hypocenter | C3 | #2880 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 | |
774 conjugacy classes of subgroups; only the 297 normal subgroups are shown.
This group has 774 conjugacy classes of subgroups — too many to draw. Only its 297 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(10,16)(13,19)(17,22)(20,23);(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23);(5,7);(2,3)(4,5)(6,7)(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23)Polycyclic presentation:
2^7,3|a^2=f,f^2=g,b^a=be,h^a=h^2,d^c=dg90 conjugacy classes — view the full character table.
| maximal subgroup of | #10691492, #10691496, #10691506, … |
| maximal normal subgroup of | #10691492, #10691496, #10691506, … |
| maximal quotient of | #10761372, #10761382, #10761391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16651);