Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#408 ⋊ C2 |
| Sylow | C3 ⋊ #3203 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#408 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #408 | D3 |
| hypocenter | C3 | #3203 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ #408 | |
402 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(8,10)(12,15,16,25)(13,19,20,23)(14,22,17,24)(18,26,21,27);(2,3)(6,11)(8,10)(13,18)(14,17)(19,26)(20,21)(22,24)(23,27);(4,6)(5,8)(7,11)(9,10)(12,14,16,17)(13,18,20,21)(15,24,25,22)(19,27,23,26);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,21)(15,23)(16,20)(17,18)(19,25)(22,26)(24,27)Polycyclic presentation:
2^7,3|b^2=g,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ce,d^b=dg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707327, #10707331, #10707344, … |
| maximal normal subgroup of | #10707327, #10707331, #10707344, … |
| maximal quotient of | #10804306, #10805502, #10806515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11252);