Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2178 |
146 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 61 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)(4,7,9,21,10,22,25,8)(5,12,13,30,15,31,14,23)(6,17,18,26,20,34,19,24)(11,27,28,33,29,35,32,16);(4,6)(5,11)(7,17)(8,24)(9,18)(10,20)(12,27)(13,28)(14,32)(15,29)(16,23)(19,25)(21,26)(22,34)(30,33)(31,35);(2,3)(4,5,9,13,10,15,25,14)(6,16,18,27,20,33,19,35)(7,12,21,30,22,31,8,23)(11,26,28,34,29,24,32,17)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,d^2=fg,e^2=g,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^2,d^b=df,f^b=fg,h^c=h^266 conjugacy classes — view the full character table.
| maximal subgroup of | #10715792, #10715795, #10715797, … |
| maximal normal subgroup of | #10715792, #10715795, #10715797, … |
| maximal quotient of | C3⋊C8×D16, C3⋊C8×SD32, C3⋊C8×Q32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4087);