Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3099 |
398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 398 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)(12,14,16,21)(13,18,20,25)(15,22,23,27)(17,24,19,26);(2,3)(4,6,7,10)(5,9,8,11)(13,17)(15,23)(18,24)(19,20)(22,27)(25,26);(5,8)(9,11)(14,21)(18,25)(22,27)(24,26);(4,5)(6,9)(7,8)(10,11)(12,13,16,20)(14,18,21,25)(15,17,23,19)(22,24,27,26)Polycyclic presentation:
2^7,3|a^2=g,c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10839415, #10839418, #10839430, … |
| maximal normal subgroup of | #10839415, #10839418, #10839430, … |
| maximal quotient of | #10795001, #10795950, #10796464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15995);