Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#575 ⋊ C2 |
| Sylow | C3 ⋊ #3020 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#937 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#575 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #575 | D3 |
| hypocenter | C3 | #3020 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #575 | |
698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,12)(10,25)(11,26)(13,20)(14,28)(24,35)(27,29)(30,31);(2,3)(4,7,9,19)(5,11,12,26)(6,15,17,33)(8,22,23,18)(10,24,25,35)(13,29,20,27)(14,30,28,31)(16,34,21,32);(4,6)(5,10)(7,15)(8,21)(9,17)(11,24)(12,25)(13,28)(14,20)(16,23)(18,34)(19,33)(22,32)(26,35)(27,30)(29,31);(4,5)(6,14)(7,11)(8,20)(9,12)(10,16)(13,23)(15,31)(17,28)(18,29)(19,26)(21,25)(22,27)(24,32)(30,33)(34,35)Polycyclic presentation:
2^7,3|c^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fg,e^c=eg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10826497, #10826498, #10826501, … |
| maximal normal subgroup of | #10826497, #10826498, #10826501, … |
| maximal quotient of | #10767569, #10767571, #10767608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16582);