Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#540 ⋊ C2 |
| Sylow | C3 ⋊ #3122 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#540 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #540 | D3 |
| hypocenter | C3 | #3122 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #540 | |
418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(14,22)(17,21)(23,27)(25,26);(2,3)(13,18)(16,24)(17,25)(19,20)(21,26)(23,27);(5,8)(9,11)(12,14)(13,17)(15,22)(16,23)(18,25)(19,21)(20,26)(24,27);(4,5)(6,9)(7,8)(10,11)(12,13)(14,21)(15,19)(16,18)(17,22)(20,24)(23,26)(25,27)Polycyclic presentation:
2^7,3|d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3122, C2×#83435, #11663549, … |
| maximal normal subgroup of | D3×#3122, C2×#83435, #11663549, … |
| maximal quotient of | #10795273, #10798272, #10798317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16011);