Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×D8 ⋊ C2 |
| Sylow | C3 ⋊ #3366 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×C4×D8 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | C4×D8 | D3 |
| hypocenter | C3 | #3366 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D8 | |
396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 396 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) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6,7,11)(5,9,8,10);(12,13)(14,19)(15,17)(16,18);(2,3)(4,5,7,8)(6,10,11,9)(13,17)(16,19);(5,8)(9,10)(13,16)(14,18)(17,19)Polycyclic presentation:
2^7,3|b^2=e,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,d^b=de,h^b=h^2,f^c=fg57 conjugacy classes — view the full character table.
| maximal subgroup of | #10785029, #10785031, #10785089, … |
| maximal normal subgroup of | #10785029, #10785031, #10785089, … |
| maximal quotient of | #10811810, #10812021, #10814162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12146);