Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #568 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×C4⋊S4 |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #568 | D3 |
| hypercenter | M16 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
257 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 257 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,21,30,7,4,24,11,19,17,2,27,32,6,9,29,18,16,26,3,25,15,8,12,31)(5,10,13,20,14,22,28,23);(2,8)(7,18)(9,21)(10,22)(19,25)(20,23)(24,26)(31,32);(2,7)(3,11)(4,14)(5,16)(8,18)(9,23)(10,25)(12,13)(15,17)(19,22)(20,21)(24,32)(26,31)(27,28)Polycyclic presentation:
2^4,3,2^3|c^2=d,d^2=h,f^2=h,g^2=h,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11713370, #11713372, #11713427, … |
| maximal normal subgroup of | #11713370, #11713372, #11713427, … |
| maximal quotient of | #11712177, #11712180, #11712181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18030);