Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2399 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C2×C3×C4×Q8 | C2 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | #2399 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×Q8 | |
504 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 177 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) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,10,11,14)(9,13,12,15)(16,19,20,18)(17,21,23,22);(5,7)(9,12)(13,15)(16,18,20,19)(17,22,23,21);(2,3)(4,5)(6,7)(16,18,20,19)(17,21,23,22);(8,9)(10,13)(11,12)(14,15)(16,17,20,23)(18,22,19,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg,h^b=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10690258, #10690259, #10690261, … |
| maximal normal subgroup of | #10690258, #10690259, #10690261, … |
| maximal quotient of | #10716928, #10716943, #10716944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15043);