Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#399 ⋊ C2 |
| Sylow | C3 ⋊ #2816 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| prime radical | C2×#884 | C2 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2816 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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)(10,14)(12,15)(16,19,20,18)(17,21,22,23);(5,7)(9,13)(12,15)(17,22)(18,19);(2,3)(4,5)(6,7)(8,10)(9,12)(11,14)(13,15)(16,18,20,19)(17,21,22,23);(8,9)(10,12)(11,13)(14,15)(16,17,20,22)(18,23,19,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,h^b=h^2,d^c=de,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10770541, #10770544, #10770563, … |
| maximal normal subgroup of | #10770541, #10770544, #10770563, … |
| maximal quotient of | #10725837, #10725839, #10725858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10666);