Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2151 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D3 |
| commutator, socle | C22×C3 | C22×C8 |
| Fitting | C3×#405 | C2 |
| prime radical | C23×C3 | C2×C8 |
| cosocle | C2×C3×C8 | C23 |
| hypercenter | #405 | D3 |
| hypocenter | C3 | #2151 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C8 ◁ #405 | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(10,18)(14,24)(19,30)(20,31)(25,34)(26,35)(32,38)(36,39);(4,5,6,7)(8,10)(9,14)(11,19)(12,20)(13,18)(15,25)(16,26)(17,24)(21,32)(22,30)(23,31)(27,36)(28,34)(29,35)(33,38)(37,39);(2,3)(5,7)(8,9,11,15,12,16,21,27,13,17,22,28,23,29,33,37)(10,14,19,25,20,26,32,36,18,24,30,34,31,35,38,39)Polycyclic presentation:
2^7,3|a^2=e,b^2=d,e^2=f,f^2=g,b^a=bd,h^a=h^2,c^b=cg96 conjugacy classes — view the full character table.
| maximal subgroup of | #10682281, #10682283, #10711321, … |
| maximal normal subgroup of | #10682281, #10682283, #10711321, … |
| maximal quotient of | #10706735, #10706761, #10706795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2458);