Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #568 ⋊ D3 |
157 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,23,32,2,5,29,12,9,21,3,27,26,7,13,31,11,19,30,4,20,18,8,15,25)(6,14,16,10,22,24,28,17);(1,2)(3,12)(4,8)(5,17)(6,20)(7,11)(9,22)(10,19)(13,16)(14,15)(18,26)(21,25)(23,28)(24,27)(29,31)(30,32)Polycyclic presentation:
2^4,3,2^3|b^2=c,c^2=h,d^2=h,f^2=h,g^2=h,b^a=bd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,d^b=dh,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73132, #11712733, #11712836, … |
| maximal normal subgroup of | C2×#73132, #11712733, #11712836, … |
| maximal quotient of | #11711108, #11711109, #11711122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5708);