Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#471 ⋊ C2 |
| Sylow | C3 ⋊ #3067 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#121 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#471 | C2 |
| socle | C2×C3 | C2×#121 |
| hypercenter | #471 | D3 |
| hypocenter | C3 | #3067 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #471 | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(2,3)(4,8,10,9)(5,13,15,14)(6,18,20,19)(7,22,23,24)(11,26,27,16)(12,28,29,21)(17,32,33,34)(25,35,31,30);(1,2,3)(4,7)(5,12)(6,17)(8,22)(9,24)(10,23)(11,25)(13,28)(14,21)(15,29)(16,30)(18,32)(19,34)(20,33)(26,35)(27,31);(4,6,8,18,10,20,9,19)(5,11,14,16,15,27,13,26)(7,17,22,32,23,33,24,34)(12,25,21,30,29,31,28,35);(4,5)(6,16)(7,21)(8,13)(9,14)(10,15)(11,18)(12,22)(17,31)(19,27)(20,26)(23,28)(24,29)(25,33)(30,32)(34,35)Polycyclic presentation:
2^7,3|b^2=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,f^c=fg,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3067, C2×#80929, #11662605, … |
| maximal normal subgroup of | D3×#3067, C2×#80929, #11662605, … |
| maximal quotient of | #10777424, #10777452, #10777495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13505);