Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×Q16 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×S4 |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C22×Q16 | D3 |
| socle | C23 | D3×D4 |
| hypercenter | Q16 | S4 |
| hypocenter | A4 | #135 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
266 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 266 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) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4)(5,7,9,13)(6,11,10,19)(8,17,18,15)(12,20,16,14);(5,6,9,10)(7,14,13,20)(8,16,18,12)(11,17,19,15);(2,3)(6,10)(7,13)(12,16)(15,17)Polycyclic presentation:
2^5,3,2^2|b^2=e,c^2=e,d^2=e,b^a=be,c^a=ce,f^a=f^2,g^a=h,h^a=g,c^b=cd,d^b=de,d^c=de,g^f=h,h^f=gh35 conjugacy classes — view the full character table.
| maximal subgroup of | #11714201, #11714203, #11714207, … |
| maximal normal subgroup of | #11714201, #11714203, #11714207, … |
| maximal quotient of | #11712345, #11712347, #11712349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18041);