Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C23×C8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4⋊S4 |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C23×C8 | D3 |
| socle | C23 | C2×D12 |
| hypercenter | C2×C8 | S4 |
| hypocenter | A4 | #135 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
273 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 38 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)(5,8,9,16,10,17,7,14)(6,12,11,15,13,19,18,20);(5,7,10,9)(6,11,13,18)(8,14,17,16)(12,15,19,20);(2,3)(5,6)(7,11)(8,15)(9,18)(10,13)(12,16)(14,19)(17,20)Polycyclic presentation:
2^5,3,2^2|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^2,g^a=h,h^a=g,g^f=h,h^f=gh44 conjugacy classes — view the full character table.
| maximal subgroup of | #11711566, #11711572, #11711600, … |
| maximal normal subgroup of | #11711566, #11711572, #11711600, … |
| maximal quotient of | #11712161, #11712165, #11712170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18016);