Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
210 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 30 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:
(5,23,28,10,15,25,11,18,21,6,17,24)(7,20,22,9,19,16,13,12,26,14,8,27);(1,2,3,4)(6,9)(7,13)(8,16)(10,14)(12,21)(15,24)(17,25)(18,22)(19,27)(20,28)(23,26)Polycyclic presentation:
2^4,3,2^3|a^2=c,b^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×#73153, #11712618, #11712619, … |
| maximal normal subgroup of | C2×#73153, #11712618, #11712619, … |
| maximal quotient of | #11711015, #11711040, #11711041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5729);