Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#578 ⋊ C2 |
| Sylow | C3 ⋊ #3290 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#578 | C2 |
| socle | C2×C3 | D42 |
| hypercenter | #578 | D3 |
| hypocenter | C3 | #3290 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #578 | |
454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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)(4,7)(5,12)(6,17)(8,23)(9,24)(10,28)(11,25)(13,32)(14,31)(15,34)(16,18)(19,21)(20,35)(22,26)(27,30)(29,33);(4,7)(5,11)(6,16)(8,22)(9,24)(10,27)(12,25)(13,31)(14,32)(15,34)(17,18)(19,35)(20,21)(23,26)(28,30)(29,33);(4,6)(5,10)(7,21)(8,19)(9,20)(11,28)(12,33)(13,29)(14,30)(15,25)(16,24)(17,22)(18,26)(23,35)(27,32)(31,34);(2,3)(4,5)(6,15)(7,12)(8,25)(9,14)(10,18)(11,23)(13,26)(16,28)(17,34)(19,30)(20,29)(21,27)(22,32)(24,31)(33,35)Polycyclic presentation:
2^7,3|e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^2,c^b=cg,d^b=de,e^b=eg,f^c=fg,e^d=eg,h^d=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10951927, #10951936, #10951942, … |
| maximal normal subgroup of | #10951927, #10951936, #10951942, … |
| maximal quotient of | #10786454, #10786479, #10786516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16596);