Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2822 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#125 | C2 |
| hypercenter | C2×#125 | D3 |
| hypocenter | C3 | #2822 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
380 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(8,11,12,15)(9,13,10,14)(18,25)(21,29)(24,31)(28,30);(5,7)(8,10,12,9)(11,13,15,14)(16,19,20,27)(17,22,23,26)(18,24,25,31)(21,28,29,30);(2,3)(4,5)(6,7)(8,9,12,10)(11,13,15,14)(16,18)(17,21)(19,24)(20,25)(22,28)(23,29)(26,30)(27,31);(9,10)(13,14)(16,17,20,23)(18,21,25,29)(19,26,27,22)(24,30,31,28)Polycyclic presentation:
2^7,3|a^2=e,b^2=ef,c^2=f,d^2=ef,b^a=bef,c^a=cf,c^b=cg,d^b=de,h^b=h^2,d^c=def54 conjugacy classes — view the full character table.
| maximal subgroup of | #10696476, #10696479, #10696502, … |
| maximal normal subgroup of | #10696476, #10696479, #10696502, … |
| maximal quotient of | #10725909, #10725910, #10725931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15631);