Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2632 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#124 | C2 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | C2×#124 | D3 |
| hypocenter | C3 | #2632 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
500 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 137 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) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,10,11,14)(9,13,12,15)(16,19,20,18)(17,21,22,23);(5,7)(9,12)(13,15)(17,22)(18,19);(2,3)(4,5)(6,7)(16,18,20,19)(17,21,22,23);(8,9)(10,13)(11,12)(14,15)(16,17,20,22)(18,23,19,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,d^2=f,b^a=be,c^a=cf,c^b=cg,h^b=h^2,d^c=de60 conjugacy classes — view the full character table.
| maximal subgroup of | #10691172, #10696075, #10696079, … |
| maximal normal subgroup of | #10691172, #10696075, #10696079, … |
| maximal quotient of | #10721658, #10721842, #10721931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15553);