Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2645 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#401 | C2 |
| hypercenter | #401 | D3 |
| hypocenter | C3 | #2645 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #401 | |
352 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 123 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)(20,24,25,21)(22,27,23,26);(5,9)(8,11)(14,18)(16,19)(20,23,25,22)(21,26,24,27);(2,3)(4,6,7,10)(5,8,9,11)(12,14)(13,16)(15,18)(17,19)(20,22)(21,26)(23,25)(24,27);(4,5,7,9)(6,8,10,11)(12,13,15,17)(14,16,18,19)(20,21,25,24)(22,27,23,26)Polycyclic presentation:
2^7,3|a^2=f,b^2=fg,c^2=e,d^2=e,b^a=be,c^a=cfg,c^b=cg,h^b=h^2,d^c=de,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2645, C2×#78416, #10962312, … |
| maximal normal subgroup of | D3×#2645, C2×#78416, #10962312, … |
| maximal quotient of | #10721591, #10721638, #10722327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 10992);