Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #5386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C4×Q8 ⋊ D3 |
224 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,3,4,7)(2,6,5,8)(9,17,22)(10,21,24)(11,12,20)(14,15,23);(2,5)(6,8)(10,14)(15,21)(16,19)(23,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,21)(15,18)(16,17)(20,24)(22,23)Polycyclic presentation:
2^4,3,2^3|c^2=dh,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 | #11711862, #11711865, #11711868, … |
| maximal normal subgroup of | #11711862, #11711865, #11711868, … |
| maximal quotient of | #11712659, #11712775, #11712777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17905);