Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting | C2×C3×#125 ⋊ C2 |
| Sylow | C3 ⋊ #2641 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×#125 | C2 |
| prime radical | C2×#874 | C2 |
| hypercenter | C2×#125 | D3 |
| hypocenter | C3 | #2641 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,18,19,22)(17,21,20,23);(2,3)(4,5)(6,7)(9,13)(12,15)(18,22)(21,23);(2,3)(4,5,6,7)(8,10,11,14)(9,12,13,15)(17,20)(21,23);(2,3)(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,20)(18,21,22,23)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,d^2=eg,b^a=beg,c^a=cf,h^a=h^2,c^b=cg,h^b=h^2,d^c=deg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10696534, #10696538, #10696590, … |
| maximal normal subgroup of | #10696534, #10696538, #10696590, … |
| maximal quotient of | #10721587, #10721711, #10721793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15643);