Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C22×C16 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #5402 |
| commutator, cosocle | C8×A4 | C22 |
| Frattini | C8 | C2×S4 |
| Fitting | C22×C16 | D3 |
| socle | C23 | D24 |
| hypercenter | C16 | S4 |
| hypocenter | A4 | D16 |
| LCS | G ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4)(5,6,11,17,8,12,15,18,9,13,19,20,14,16,7,10);(2,3)(6,10)(7,11)(8,14)(12,20)(13,18)(15,19)(16,17)Polycyclic presentation:
2^5,3,2^2|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=h,h^a=g,g^f=h,h^f=gh38 conjugacy classes — view the full character table.
| maximal subgroup of | #11711662, #11711663, #11711675, … |
| maximal normal subgroup of | #11711662, #11711663, #11711675, … |
| maximal quotient of | #11711470, #11711473, #11711476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5611);