Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#473 ⋊ C2 |
| Sylow | C3 ⋊ #3293 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #5609 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#473 | C2 |
| socle | C2×C3 | D42 |
| hypercenter | #473 | D3 |
| hypocenter | C3 | #3293 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #473 | |
444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 444 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)(4,8)(5,13)(6,18)(7,21)(9,27)(10,24)(11,25)(12,28)(14,32)(15,30)(16,31)(17,33)(19,34)(20,35)(22,29)(23,26);(2,3)(4,7,11,12)(5,6,16,20)(8,21,25,28)(9,22,10,17)(13,18,31,35)(14,19,15,26)(23,32,34,30)(24,33,27,29);(2,3)(4,6,11,20)(5,12,16,7)(8,23,25,34)(9,26,10,19)(13,29,31,33)(14,22,15,17)(18,27,35,24)(21,30,28,32);(2,3)(4,5,11,16)(6,17,20,22)(7,19,12,26)(8,13,25,31)(9,15,10,14)(18,33,35,29)(21,34,28,23)(24,32,27,30)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=g,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,h^b=h^2,f^c=fg,h^c=h^2,e^d=eg45 conjugacy classes — view the full character table.
| maximal subgroup of | #10951945, #10951946, #10951952, … |
| maximal normal subgroup of | #10951945, #10951946, #10951952, … |
| maximal quotient of | #10786495, #10786497, #10786503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13523);