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 |
165 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 165 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:
(1,19,28,8,4,23,11,21,17,2,25,32,6,9,27,18,16,24,3,29,15,7,12,31)(5,10,13,22,14,20,26,30);(2,7)(3,11)(4,14)(5,16)(9,10)(12,13)(15,17)(19,20)(21,30)(22,29)(25,26)(31,32)Polycyclic presentation:
2^4,3,2^3|b^2=c,c^2=h,f^2=h,g^2=h,b^a=bd,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,d^b=dh,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73131, #11712759, #11712948, … |
| maximal normal subgroup of | C2×#73131, #11712759, #11712948, … |
| maximal quotient of | #11711149, #11711167, #11711171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5707);