Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2569 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×C4×C22⋊C4 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | C4×C22⋊C4 | D3 |
| hypocenter | C3 | #2569 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
456 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 153 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)(5,9)(8,10);(2,3)(14,18)(16,19)(20,22,23,27)(21,24,25,26);(4,6,7,11)(5,8,9,10)(12,14,15,18)(13,16,17,19);(4,5)(6,10)(7,9)(8,11)(12,13,15,17)(14,16,18,19)(20,21,23,25)(22,26,27,24)Polycyclic presentation:
2^7,3|a^2=f,b^2=eg,c^2=fg,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 | #10698349, #10698350, #10698356, … |
| maximal normal subgroup of | #10698349, #10698350, #10698356, … |
| maximal quotient of | #10717443, #10717950, #10718092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6115);