Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2276 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C2×C3 | C22×C42 |
| Fitting | C3×C4×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C4×C4⋊C4 | D3 |
| hypocenter | C3 | #2276 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
628 conjugacy classes of subgroups; only the 325 normal subgroups are shown.
This group has 628 conjugacy classes of subgroups — too many to draw. Only its 325 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(12,15,16,13)(14,18,19,17);(8,10,11,9)(12,14,16,19)(13,17,15,18);(2,3)(4,5,6,7)(12,13,16,15)(14,18,19,17);(2,3)(8,9,11,10)(13,15)(17,18)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=ef,d^2=e,b^a=be,d^a=de,h^a=h^2,c^b=ce,h^b=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #10692088, #10692092, #10692102, … |
| maximal normal subgroup of | #10692088, #10692092, #10692102, … |
| maximal quotient of | #10716271, #10716272, #10716473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6192);