Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C3×#585 ⋊ C2 |
| Sylow | C3 ⋊ 2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D3 |
| commutator, socle, cosocle | C2×C3 | C26 |
| Fitting | C3×#585 | C2 |
| hypercenter | #585 | D3 |
| hypocenter | C3 | 2−1+6 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #585 | |
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 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 — 6 generators on 35 points:
(1,2,3)(5,13)(10,14)(11,25)(12,27)(22,28)(23,29)(24,26)(34,35);(6,17)(10,14)(15,31)(16,33)(22,28)(23,29)(30,32)(34,35);(4,8,9,21)(5,12,13,27)(6,16,17,33)(7,18,19,20)(10,23,14,29)(11,24,25,26)(15,30,31,32)(22,34,28,35);(2,3)(4,7,9,19)(5,11,13,25)(6,15,17,31)(8,20,21,18)(10,22,14,28)(12,26,27,24)(16,32,33,30)(23,35,29,34);(4,6)(5,10)(7,15)(8,16)(9,17)(11,22)(12,23)(13,14)(18,30)(19,31)(20,32)(21,33)(24,34)(25,28)(26,35)(27,29);(4,5)(6,14)(7,11)(8,12)(9,13)(10,17)(15,28)(16,29)(18,24)(19,25)(20,26)(21,27)(22,31)(23,33)(30,35)(32,34)Polycyclic presentation:
2^7,3|c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg,h^c=h^299 conjugacy classes — view the full character table.
| maximal subgroup of | #10943935, #10943937, #10943939, … |
| maximal normal subgroup of | #10943935, #10943937, #10943939, … |
| maximal quotient of | #11701807, #11701809, #11701811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20148);