Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#401 ⋊ C2 |
| Sylow | C3 ⋊ #2834 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#401 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | #401 | D3 |
| hypocenter | C3 | #2834 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #401 | |
372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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)(12,14)(13,17)(15,19)(16,18)(21,25)(24,27);(6,10)(8,11)(14,19)(16,17)(20,22,23,26)(21,24,25,27);(4,6)(5,8)(7,10)(9,11)(12,14,15,19)(13,16,18,17)(20,21)(22,24)(23,25)(26,27);(2,3)(4,5,7,9)(6,8,10,11)(12,13)(14,17)(15,18)(16,19)(22,26)(24,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,c^2=f,b^a=beg,c^a=cf,h^a=h^2,c^b=cg,d^b=defg,d^c=deg,h^d=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2834, C2×#78438, #11001481, … |
| maximal normal subgroup of | D3×#2834, C2×#78438, #11001481, … |
| maximal quotient of | #10725893, #10725981, #10726067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11014);