Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
199 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 199 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,3,4,7)(2,5,6,8)(9,24,26)(10,17,32,16,25,31)(11,35,37)(12,23,14)(15,29,40,28,36,39)(18,22)(19,34,21)(30,33);(3,7)(5,8)(9,11,14,21)(10,15,16,28)(12,19,24,35)(13,20,27,38)(17,29,25,36)(18,30,22,33)(23,34,26,37)(31,39,32,40);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,22)(13,25)(14,16)(17,27)(18,24)(19,33)(20,36)(21,28)(23,32)(26,31)(29,38)(30,35)(34,40)(37,39)Polycyclic presentation:
2^4,3,2^3|a^2=d,b^2=h,c^2=d,f^2=h,g^2=h,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85405, #11715018, #11715028, … |
| maximal normal subgroup of | C2×#85405, #11715018, #11715028, … |
| maximal quotient of | #11712909, #11712986, #11712995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17981);