Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×GL(2,ℤ/4) |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C2×#67 |
| hypercenter | #41 | S4 |
| hypocenter | A4 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #41 | |
336 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 336 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) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,4)(5,6)(7,12)(8,10)(9,11);(6,10)(9,12);(2,3)(6,9)(7,11)(10,12)Polycyclic presentation:
2^5,3,2^2|d^2=e,c^a=cd,d^a=de,f^a=f^2,g^a=h,h^a=g,c^b=ce,d^c=de,g^f=h,h^f=gh32 conjugacy classes — view the full character table.
| maximal subgroup of | #11713353, #11713359, #11713415, … |
| maximal normal subgroup of | #11713353, #11713359, #11713415, … |
| maximal quotient of | #11711988, #11711993, #11711996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18089);