Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C23×C8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×S4 |
| commutator | C2×A4 | C22×C4 |
| Frattini | C4 | C22×S4 |
| Fitting | C23×C8 | D3 |
| socle | C23 | C2×C4×D3 |
| prime radical | #5912 | C2 |
| cosocle | C4×A4 | C23 |
| hypercenter | C2×C8 | S4 |
| hypocenter | A4 | #131 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
273 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4)(5,7,8,13,9,14,15,19)(6,11,12,18,10,16,17,20);(6,10)(11,16)(12,17)(18,20);(2,3)(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)Polycyclic presentation:
2^5,3,2^2|c^2=d,d^2=e,b^a=be,f^a=f^2,g^a=h,h^a=g,g^f=h,h^f=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #11711559, #11711562, #11712821, … |
| maximal normal subgroup of | #11711559, #11711562, #11712821, … |
| maximal quotient of | #11711856, #11711857, #11711858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18013);