Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2980 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting, prime radical | C3×#532 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | #2980 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
394 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 197 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)(12,14)(13,18)(15,21)(16,20)(17,23)(19,25)(22,26)(24,27);(6,11)(8,10)(13,17)(14,20)(19,24)(21,26);(4,6)(5,8)(7,11)(9,10)(14,20)(18,23)(21,26)(25,27);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13,15,19,16,17,22,24)(14,18,21,25,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=f,f^2=g,b^a=be,c^a=cg,d^a=de,h^a=h^2,c^b=ce,d^b=dg,d^c=dg66 conjugacy classes — view the full character table.
| maximal subgroup of | #10689589, #10689592, #10690134, … |
| maximal normal subgroup of | #10689589, #10689592, #10690134, … |
| maximal quotient of | #10793914, #10793951, #10794170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15855);