Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3082 |
362 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(13,18)(17,25)(19,26)(24,27);(2,3)(12,14,16,21)(13,17,18,25)(15,23,22,20)(19,27,26,24);(2,3)(4,6,7,11)(5,8,9,10)(12,13)(14,17)(15,19)(16,18)(20,24)(21,25)(22,26)(23,27);(4,5,7,9)(6,10,11,8)(14,20)(15,22)(17,24)(19,26)(21,23)(25,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=e,c^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,h^b=h^2,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10858635, #10858640, #10858665, … |
| maximal normal subgroup of | #10858635, #10858640, #10858665, … |
| maximal quotient of | #10794704, #10796178, #10796212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11822);