Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#448 ⋊ C2 |
| Sylow | C3 ⋊ #3326 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#448 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #448 | D3 |
| hypocenter | C3 | #3326 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #448 | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,11)(8,10)(13,18)(17,26)(19,23)(20,27);(12,14,16,21)(13,17,18,26)(15,24,25,22)(19,27,23,20);(2,3)(4,6)(5,8)(7,11)(9,10)(14,21)(17,26)(20,27)(22,24);(2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,20)(15,23)(16,18)(17,22)(19,25)(21,27)(24,26)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,f^c=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3326, C2×#80055, #11672137, … |
| maximal normal subgroup of | D3×#3326, C2×#80055, #11672137, … |
| maximal quotient of | #10810817, #10811624, #10812429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12631);