Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C4×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #2496 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×C4×C4⋊C4 | C2 |
| prime radical | C23×D3 | C23 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2496 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(4,6,7,5);(2,3)(9,13)(12,15)(18,22)(20,23);(4,5,7,6)(8,10,11,14)(9,12,13,15)(16,18)(17,20)(19,22)(21,23);(5,6)(8,9,11,13)(10,12,14,15)(16,17,19,21)(18,20,22,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=efg,d^2=e,b^a=be,c^a=cfg,d^a=de,c^b=cg,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10867844, #10867848, #10867904, … |
| maximal normal subgroup of | #10867844, #10867848, #10867904, … |
| maximal quotient of | #10717132, #10717568, #10718030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6368);