Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×S4 |
| commutator, prime radical, cosocle | C22×A4 | C23 |
| Fitting | C22×C4⋊C4 | D3 |
| socle | C24 | C22×D3 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | A4 | #125 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
255 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 96 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4)(6,10)(9,12)(13,15,16,19)(14,17,18,20);(5,7,8,11)(6,9,10,12)(15,19)(17,20);(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,19,20)Polycyclic presentation:
2^5,3,2^2|a^2=e,b^2=d,c^2=e,c^a=cd,f^a=f^2,g^a=h,h^a=g,c^b=ce,g^f=h,h^f=gh38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85304, #11714338, #11714347, … |
| maximal normal subgroup of | C2×#85304, #11714338, #11714347, … |
| maximal quotient of | #11711703, #11711705, #11711718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17880);