Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Fitting | C3×#396 ⋊ C2 |
| Sylow | C3 ⋊ #2825 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | #2825 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
522 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 522 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) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,10)(8,11)(22,27)(25,26);(2,3)(5,9)(8,11)(14,18)(16,19)(20,22)(21,25)(23,27)(24,26);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14)(13,16)(15,18)(17,19)(21,24)(22,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,16,18,19)(20,21)(22,26)(23,24)(25,27)Polycyclic presentation:
2^7,3|a^2=e,b^a=beg,c^a=cf,h^a=h^2,c^b=cg,d^b=defg,h^b=h^2,d^c=deg,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2825, C2×#77451, #11001578, … |
| maximal normal subgroup of | D3×#2825, C2×#77451, #11001578, … |
| maximal quotient of | #10725828, #10725987, #10726039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10027);