Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2817 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C22⋊Q8 | C2 |
| prime radical | C24×C3 | C23 |
| hypercenter | C2×C22⋊Q8 | D3 |
| hypocenter | C3 | #2817 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
444 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 444 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) = 192 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(10,14)(12,15)(16,20,21,18)(17,22,24,25)(19,26,29,28)(23,31,27,30);(5,7)(9,13)(12,15)(16,19,21,29)(17,23,24,27)(18,26,20,28)(22,30,25,31);(2,3)(4,5)(6,7)(8,10)(9,12)(11,14)(13,15)(16,18,21,20)(17,22,24,25)(19,28,29,26)(23,31,27,30);(8,9)(10,12)(11,13)(14,15)(16,17,21,24)(18,25,20,22)(19,27,29,23)(26,30,28,31)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,h^b=h^2,d^c=de54 conjugacy classes — view the full character table.
| maximal subgroup of | #10695830, #10695832, #10695847, … |
| maximal normal subgroup of | #10695830, #10695832, #10695847, … |
| maximal quotient of | #10725878, #10725879, #10725889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15384);