Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#433 ⋊ C2 |
| Sylow | C3 ⋊ #3223 |
364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(5,9)(8,10)(14,23)(18,32)(22,37)(24,39)(31,42)(33,40)(36,43)(38,41);(2,3)(6,11)(8,10)(12,15,17,27)(13,19,21,34)(14,22,23,37)(16,29,30,26)(18,31,32,42)(20,35,28,25)(24,41,39,38)(33,43,40,36);(4,6,7,11)(5,8,9,10)(12,14)(13,18)(15,22)(16,24)(17,23)(19,31)(20,33)(21,32)(25,36)(26,38)(27,37)(28,40)(29,41)(30,39)(34,42)(35,43);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13,17,21)(14,18,23,32)(15,25,27,35)(16,28,30,20)(19,29,34,26)(22,36,37,43)(24,40,39,33)(31,41,42,38)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ce,d^b=dg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3223, C2×#79398, #11666590, … |
| maximal normal subgroup of | D3×#3223, C2×#79398, #11666590, … |
| maximal quotient of | #10804574, #10804978, #10805787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11974);