Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #5386 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊GU(2,𝔽3) |
144 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,15,28)(6,19,27)(7,20,16)(8,26,13)(9,23,25)(10,18,24)(11,12,17)(14,21,22);(2,4)(5,6,7,11,10,14,13,9)(8,16,15,24,20,26,18,28)(12,22,21,25,23,27,19,17)Polycyclic presentation:
2^4,3,2^3|a^2=c,b^2=d,c^2=h,f^2=h,g^2=h,b^a=bd,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #11711041, #11711042, #11711043, … |
| maximal normal subgroup of | #11711041, #11711042, #11711043, … |
| maximal quotient of | C4⋊(SL(2,𝔽3)⋊C8), #11711065, #11711076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5646);