Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C3 ⋊ #2366 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C2×C3×#127 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#127 | D3 |
| hypocenter | C3 | #2366 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
832 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 832 conjugacy classes of subgroups — too many to draw. Only its 261 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(12,14,15,13);(8,9,10,11);(2,3)(4,6,7,5)(13,14);(2,3)(4,5,7,6)(9,11)(12,13,15,14)Polycyclic presentation:
2^7,3|a^2=g,b^2=eg,c^2=f,d^2=e,b^a=be,c^a=cf,h^a=h^2,d^b=de,h^b=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10689599, #10689602, #10689604, … |
| maximal normal subgroup of | #10689599, #10689602, #10689604, … |
| maximal quotient of | #10716627, #10716756, #10716791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15749);