Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
192 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,17,47,15,32,42,7,23,34,3,20,19)(2,27,45,12,46,39,10,44,36,8,30,43)(4,5,41,13,35,29,6,21,48,16,14,40)(9,11,18,24,31,33,22,28,38,26,25,37);(1,2,4,9,7,10,6,22)(3,12,13,26,15,8,16,24)(5,18,17,36,21,38,23,45)(11,29,27,42,28,40,44,19)(14,33,32,39,35,37,20,43)(25,41,46,34,31,48,30,47)Polycyclic presentation:
2^4,3,2^3|a^2=c,b^2=h,c^2=h,f^2=h,g^2=h,b^a=bd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11711511, #11711513, #11711619, … |
| maximal normal subgroup of | #11711511, #11711513, #11711619, … |
| maximal quotient of | #11711018, #11711027, #11711033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5728);