Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#399 ⋊ C2 |
| Sylow | C3 ⋊ #2796 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#399 | C2 |
| prime radical | C22×#67 | C22 |
| hypercenter | #399 | D3 |
| hypocenter | C3 | #2796 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #399 | |
434 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)(5,9)(8,11)(13,17)(14,19)(21,25)(24,26);(2,3)(5,9)(6,10)(14,19)(16,18)(20,22,23,27)(21,24,25,26);(2,3)(4,6)(5,8)(7,10)(9,11)(12,14,15,19)(13,16,17,18);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,18,19,16)(20,21)(22,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=efg,b^a=be,c^a=cf,d^a=def,h^a=h^2,c^b=cg,d^b=de,h^b=h^2,h^c=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #11003016, #11003035, #11003173, … |
| maximal normal subgroup of | #11003016, #11003035, #11003173, … |
| maximal quotient of | #10723445, #10723460, #10723782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10700);