Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived | SL(2,𝔽3) ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C42 |
| Fitting | C2×C4×Q8 | D3 |
| prime radical, cosocle | C22×SL(2,𝔽3) | C22 |
| hypercenter | C22×C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
177 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(5,6,7,8)(9,19,21)(10,15,24,14,20,23)(11,18,13)(16,17);(1,2,3,4)(9,10)(11,17)(12,20)(13,14)(15,22)(16,19)(18,24)(21,23)Polycyclic presentation:
2^4,3,2^3|a^2=c,b^2=d,f^2=h,g^2=h,b^a=bh,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #11711039, #11711040, #11711168, … |
| maximal normal subgroup of | #11711039, #11711040, #11711168, … |
| maximal quotient of | #11710987, #11710988, #11711000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5639);