Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1840 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator, socle, prime radical | C23×C3 | C22×C4 |
| Fitting | C3×#336 | C2 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | #336 | D3 |
| hypocenter | C3 | #1840 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ #336 | |
196 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(5,9)(8,11)(12,14,15,19)(13,17,16,18)(22,28)(25,31)(29,34)(32,35);(2,3)(4,6,7,10)(5,8,9,11)(13,16)(17,18)(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35);(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,16)(17,19)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35)Polycyclic presentation:
2^7,3|a^2=f,b^2=de,c^2=d,f^2=g,b^a=bd,c^a=ce,c^b=cg,h^b=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1840, C2×#68608, #10795438, … |
| maximal normal subgroup of | D3×#1840, C2×#68608, #10795438, … |
| maximal quotient of | #10688912, #10688913, #10688914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1184);