Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2992 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#529 | C2 |
| prime radical | C3×#121 | C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #529 | D3 |
| hypocenter | C3 | #2992 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #529 | |
356 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)(4,6,7,10)(5,9,8,11)(14,20)(17,23)(21,26)(24,27);(5,8)(9,11);(4,5)(6,9)(7,8)(10,11)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(2,3)(5,8)(6,10)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=f,d^2=e,f^2=g,b^a=be,d^a=de,h^a=h^2,c^b=ce,d^b=dg72 conjugacy classes — view the full character table.
| maximal subgroup of | #10689303, #10689307, #10689310, … |
| maximal normal subgroup of | #10689303, #10689307, #10689310, … |
| maximal quotient of | #10793985, #10794056, #10794131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15706);