Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2310 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C23×C3×Q8 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | #2310 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
784 conjugacy classes of subgroups; only the 309 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 309 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) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(13,19)(17,20)(18,27)(25,26);(9,11)(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26);(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25);(2,3)(4,5,6,7)(8,9)(10,11)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^2,c^b=ce84 conjugacy classes — view the full character table.
| maximal subgroup of | #10688707, #10688710, #10692919, … |
| maximal normal subgroup of | #10688707, #10688710, #10692919, … |
| maximal quotient of | #10716433, #10716503, #10716577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16638);