Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#455 ⋊ C2 |
| Sylow | C3 ⋊ #3289 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #5590 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#455 | C2 |
| socle | C2×C3 | D42 |
| hypercenter | #455 | D3 |
| hypocenter | C3 | #3289 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #455 | |
448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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)(4,7)(5,13)(6,19)(8,25)(9,22)(10,23)(11,27)(12,14)(15,29)(16,31)(17,33)(18,24)(20,35)(21,34)(26,28)(30,32);(4,7)(5,12)(6,18)(8,22)(9,25)(10,23)(11,26)(13,14)(15,31)(16,29)(17,33)(19,24)(20,34)(21,35)(27,28)(30,32);(2,3)(4,6,10,21)(5,11,16,28)(7,18,23,35)(8,24,9,20)(12,26,29,27)(13,30,31,33)(14,32,15,17)(19,25,34,22);(2,3)(4,5,10,16)(6,17,21,32)(7,13,23,31)(8,15,9,14)(11,24,28,20)(12,25,29,22)(18,26,35,27)(19,33,34,30)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^2,d^b=de,e^b=eg,h^b=h^2,f^c=fg,e^d=eg42 conjugacy classes — view the full character table.
| maximal subgroup of | #10715924, #10715928, #10715938, … |
| maximal normal subgroup of | #10715924, #10715928, #10715938, … |
| maximal quotient of | #10784568, #10784577, #10784602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12871);