Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3261 |
322 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 109 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)(4,6,7,10)(5,9,8,11)(14,23)(18,32)(22,37)(24,39)(31,42)(33,40)(36,43)(38,41);(5,8)(9,11)(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);(4,5)(6,9)(7,8)(10,11)(12,14,17,23)(13,18,21,32)(15,22,27,37)(16,24,30,39)(19,31,34,42)(20,33,28,40)(25,36,35,43)(26,38,29,41);(2,3)(5,8)(6,10)(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=g,c^2=g,d^2=e,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^d=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | #10707699, #10707702, #10708549, … |
| maximal normal subgroup of | #10707699, #10707702, #10708549, … |
| maximal quotient of | #10805180, #10805611, #10807399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13474);