Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C2×C3×#124 ⋊ C2 |
| Sylow | C3 ⋊ #2631 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#124 | C2 |
| hypercenter | C2×#124 | D3 |
| hypocenter | C3 | #2631 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
628 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 628 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(10,14)(13,15)(16,18,19,22)(17,21,20,23);(5,7)(8,10)(9,13)(11,14)(12,15)(17,20)(21,23);(2,3)(4,5)(6,7)(9,12)(13,15);(8,9,11,12)(10,13,14,15)(16,17)(18,21)(19,20)(22,23)Polycyclic presentation:
2^7,3|a^2=e,d^2=f,b^a=be,c^a=cf,c^b=cg,h^b=h^2,d^c=de60 conjugacy classes — view the full character table.
| maximal subgroup of | #10691167, #10696061, #10696065, … |
| maximal normal subgroup of | #10691167, #10696061, #10696065, … |
| maximal quotient of | #10721643, #10721842, #10721931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15578);