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 ⋊ #3339 |
| 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 | #3339 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D8 | |
366 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 115 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)(6,11)(9,10)(14,22)(17,21)(23,27)(25,26);(4,7,8,5)(6,10,11,9)(13,18)(15,24)(17,25)(19,20)(21,26)(23,27);(2,3)(4,6)(5,9)(7,10)(8,11)(12,14,16,22)(13,17,20,21)(15,23,24,27)(18,25,19,26);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,21)(15,18)(16,20)(17,22)(19,24)(23,26)(25,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=e,f^2=g,b^a=beg,c^a=cf,f^a=fg,h^a=h^2,d^b=deg,h^b=h^2,f^c=fg,h^d=h^263 conjugacy classes — view the full character table.
| maximal subgroup of | #10952052, #10952054, #10952104, … |
| maximal normal subgroup of | #10952052, #10952054, #10952104, … |
| maximal quotient of | #10811443, #10812835, #10812851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12191);