Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C22×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C23×D3 |
| hypercenter | #41 | S4 |
| hypocenter | A4 | 2−1+4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ #41 | |
772 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 4 generators on 20 points:
(1,2,4)(5,9,10,8)(6,12,13,17)(7,14,15,16)(11,19,20,18);(5,8,10,9)(6,12,13,17)(7,14,15,16)(11,18,20,19);(5,7,10,15)(6,11,13,20)(8,16,9,14)(12,19,17,18);(2,3)(5,6)(7,11)(8,12)(9,17)(10,13)(14,18)(15,20)(16,19)Polycyclic presentation:
2^5,3,2^2|b^2=e,c^2=e,d^2=e,d^a=de,f^a=f^2,g^a=h,h^a=g,c^b=ce,g^f=h,h^f=gh44 conjugacy classes — view the full character table.
| maximal subgroup of | #11713181, #11713183, #11713417, … |
| maximal normal subgroup of | #11713181, #11713183, #11713417, … |
| maximal quotient of | #11714313, #11714316, #11714317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20066);