Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2829 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting, prime radical | C3×#392 | C2 |
| hypercenter | #392 | D3 |
| hypocenter | C3 | #2829 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #392 | |
450 conjugacy classes of subgroups.
| 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)(20,22,23,26)(21,25,24,27);(5,9)(8,11)(14,18)(16,19)(22,26)(25,27);(4,6)(5,8)(7,10)(9,11)(12,14)(13,16)(15,18)(17,19)(21,24)(22,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,16,18,19)(20,21,23,24)(22,25,26,27)Polycyclic presentation:
2^7,3|a^2=e,d^2=eg,b^a=beg,c^a=cf,h^a=h^2,c^b=cg,d^b=defg,d^c=deg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2829, C2×#76751, #11001581, … |
| maximal normal subgroup of | D3×#2829, C2×#76751, #11001581, … |
| maximal quotient of | #10725849, #10725863, #10725959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9327);