Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2665 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×C4×C22⋊C4 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | C4×C22⋊C4 | D3 |
| hypocenter | C3 | #2665 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
424 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,10)(9,11)(12,14,15,18)(13,17,16,19);(2,3)(4,6)(5,9)(7,10)(8,11)(13,16)(17,19)(20,22,23,27)(21,24,25,26);(5,8)(9,11)(20,21,23,25)(22,26,27,24);(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|b^2=g,c^2=g,d^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10698393, #10698394, #10698400, … |
| maximal normal subgroup of | #10698393, #10698394, #10698400, … |
| maximal quotient of | #10721796, #10721805, #10721905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6133);