Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×H(3,𝔽4) ⋊ C2 |
| Sylow | C3 ⋊ #3529 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D3 |
| hypocenter | C3 | #3529 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(12,15,16,23)(13,19,18,25)(14,22,21,27)(17,24,20,26);(5,9)(8,11)(13,18)(14,21)(19,25)(22,27);(5,9)(6,10)(14,21)(17,20)(22,27)(24,26);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14)(13,17)(15,22)(16,21)(18,20)(19,24)(23,27)(25,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,20)(15,19)(16,18)(17,21)(22,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|e^2=f,b^a=bf,c^a=cg,d^a=dfg,h^a=h^2,c^b=cfg,d^b=df,h^b=h^2,h^e=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10776336, #10776342, #10776361, … |
| maximal normal subgroup of | #10776336, #10776342, #10776361, … |
| maximal quotient of | #10957797, #10957825, #10957858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19661);