Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2170 |
146 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(4,7,9,16,8,21,23,27,10,22,25,29,24,35,28,11)(5,6,13,18,14,19,31,33,15,20,32,34,12,17,30,26);(4,6)(5,11)(7,13)(8,19)(9,18)(10,20)(12,29)(14,16)(15,27)(17,24)(21,31)(22,32)(23,33)(25,34)(26,28)(30,35);(2,3)(4,5,9,13,8,14,23,31,10,15,25,32,24,12,28,30)(6,16,18,21,19,27,33,22,20,29,34,35,17,11,26,7)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,d^a=dg,h^a=h^2,d^b=dg,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10706940, #10706943, #10706945, … |
| maximal normal subgroup of | #10706940, #10706943, #10706945, … |
| maximal quotient of | D8×C3⋊C16, #10706904, SD16×C3⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3499);