Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C3×#131 ⋊ C2 |
| Sylow | C3 ⋊ #3573 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C23×D3 |
| commutator, socle | C2×C3 | C24×C4 |
| Frattini | C4 | C24×D3 |
| Fitting | C2×C3×#131 | C2 |
| prime radical | #5964 | C2 |
| cosocle | C3×C4 | C25 |
| hypercenter | C2×#131 | D3 |
| hypocenter | C3 | #3573 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ C2×#131 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(6,15)(10,21)(14,26)(16,28)(20,30)(22,32)(27,34)(31,35);(7,17)(11,23)(14,26)(18,29)(20,30)(24,33)(27,34)(31,35);(4,7)(5,11)(6,14)(8,18)(9,17)(10,20)(12,24)(13,23)(15,26)(16,27)(19,29)(21,30)(22,31)(25,33)(28,34)(32,35);(2,3)(4,6)(5,10)(7,14)(8,16)(9,15)(11,20)(12,22)(13,21)(17,26)(18,27)(19,28)(23,30)(24,31)(25,32)(29,34)(33,35);(4,5,8,12,9,13,19,25)(6,10,16,22,15,21,28,32)(7,11,18,24,17,23,29,33)(14,20,27,31,26,30,34,35)Polycyclic presentation:
2^7,3|a^2=f,f^2=g,e^b=eg,h^b=h^2,d^c=dg108 conjugacy classes — view the full character table.
| maximal subgroup of | #10710634, #10710636, #10710637, … |
| maximal normal subgroup of | #10710634, #10710636, #10710637, … |
| maximal quotient of | #11658605, #11658606, #11658607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19736);