Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C3×#497 ⋊ C2 |
| Sylow | C3 ⋊ #3150 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#497 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #497 | D3 |
| hypocenter | C3 | #3150 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #497 | |
564 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 564 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)(8,11,13,12)(9,15,16,21)(10,19,18,20)(14,17,22,23);(4,5,6,7)(10,18)(14,22)(17,23)(19,20);(8,10,12,20,13,18,11,19)(9,14,15,17,16,22,21,23);(2,3)(5,7)(8,9)(10,17)(11,15)(12,21)(13,16)(14,20)(18,23)(19,22)Polycyclic presentation:
2^7,3|b^2=e,c^2=f,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,c^b=cg,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10715474, #10776342, #10776344, … |
| maximal normal subgroup of | #10715474, #10776342, #10776344, … |
| maximal quotient of | #10782916, #10782922, #10782957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14445);