Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2158 |
152 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 152 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) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,11,14,13)(9,16,18,28)(10,21,24,23)(12,25,27,20)(15,29,30,33)(17,26,31,39)(19,32,34,38)(22,36,37,35);(2,3)(8,10,20,35,13,23,27,37,14,24,25,36,11,21,12,22)(9,15,26,32,16,29,31,34,18,30,39,38,28,33,17,19);(4,5,6,7)(8,9)(10,19)(11,16)(12,26)(13,28)(14,18)(15,22)(17,20)(21,32)(23,38)(24,34)(25,31)(27,39)(29,36)(30,37)(33,35)Polycyclic presentation:
2^7,3|a^2=e,b^2=df,c^2=g,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10710575, #10710577, #10710580, … |
| maximal normal subgroup of | #10710575, #10710577, #10710580, … |
| maximal quotient of | #10706770, #10706834, #10706875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3224);