Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C4×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | C4×#41 | D3 |
| hypercenter | C42 | S4 |
| hypocenter | SL(2,𝔽3) | #41 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C42 | |
229 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 34 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)(9,15,21,10,19,23,12,17,22,11,13,24)(14,18,20,16);(2,5)(6,8)(9,10,12,11)(13,15,19,17)(14,16,20,18)(21,23,22,24);(1,2)(3,6)(4,5)(7,8)(10,11)(13,20)(14,19)(15,16)(17,18)(21,22)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #11711167, #11711202, #11712002, … |
| maximal normal subgroup of | #11711167, #11711202, #11712002, … |
| maximal quotient of | #11712665, #11712779, #11712811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17924);