Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C4⋊C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×S4 |
| commutator, cosocle | C22×A4 | C23 |
| Fitting | C22×C4⋊C4 | D3 |
| socle | C24 | C22×D3 |
| prime radical | C2×C4⋊S4 | C2 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | A4 | #123 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
354 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4)(5,6,8)(10,14)(13,16);(2,4)(9,11,12,15)(10,13,14,16);(6,7)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^5,3,2^2|b^2=d,c^2=e,c^a=cd,f^a=f^2,g^a=h,h^a=g,c^b=ce,g^f=h,h^f=gh38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85309, #11714331, #11714352, … |
| maximal normal subgroup of | C2×#85309, #11714331, #11714352, … |
| maximal quotient of | #11711703, #11711704, #11711714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17885);