Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2402 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×#376 | C2 |
| hypercenter | #376 | D3 |
| hypocenter | C3 | #2402 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
400 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 157 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) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(12,14,15,18)(13,17,16,19)(20,23,24,22)(21,25,26,27);(2,3)(6,10)(8,11)(20,22,24,23)(21,25,26,27);(4,6)(5,8)(7,10)(9,11)(13,16)(17,19)(20,22,24,23)(21,25,26,27);(4,5,7,9)(6,8,10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,27)(23,25)(24,26)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10698325, #10698326, #10698330, … |
| maximal normal subgroup of | #10698325, #10698326, #10698330, … |
| maximal quotient of | #10716910, #10716931, #10716963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5974);