Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #3358⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #3363⋊C3 |
| Fitting, Sylow | #3363 ⋊ C3 |
106 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(19,25,20)(22,29,23)(26,28,27)(30,32,31);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,29)(18,26)(19,23)(20,31)(21,30)(22,28)(24,25)(27,32)Polycyclic presentation:
2^2,3,2^5|a^2=eh,b^2=e,d^2=e,f^2=h,g^2=h,b^a=bd,d^a=de,d^b=de,f^c=fgh,g^c=f,g^f=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3363⋊C3, #61303⋊C3, #61319⋊C3, … |
| maximal normal subgroup of | C2×#3363⋊C3, #61303⋊C3, #61319⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×#103, #22542⋊C3, #22740⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5842);