Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #2327 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C2×C3×C4×Q8 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | #2327 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
516 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 205 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)(12,15,16,24)(13,19,18,23)(14,22,21,27)(17,25,20,26);(2,3)(8,9,10,11)(13,18)(14,21)(19,23)(22,27);(4,6,7,5)(12,14)(13,17)(15,22)(16,21)(18,20)(19,25)(23,26)(24,27);(4,5,7,6)(9,11)(12,13)(14,20)(15,23)(16,18)(17,21)(19,24)(22,25)(26,27)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10690684, #10690685, #10690687, … |
| maximal normal subgroup of | #10690684, #10690685, #10690687, … |
| maximal quotient of | #10716446, #10716498, #10716503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15032);