Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C4⋊#994 |
| unambiguous semidirect product | #3498⋊C3 |
| Fitting, Sylow | #3498 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3498 | C3 |
| hypercenter | C4⋊Q8 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
393 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,14,26,15,8,23,10,21,16,7,18,24)(6,19,25,12,11,28,13,17,20,9,22,27);(1,2,3,4)(5,13,10,6)(7,12,15,9)(8,14,18,21)(11,17,22,19)(16,27,26,28)(20,24,25,23);(2,4)(5,6,10,13)(7,12,15,9)(8,11,18,22)(14,17,21,19)(16,20,26,25)(23,27,24,28)Polycyclic presentation:
2^3,3,2^4|a^2=h,b^2=e,c^2=h,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=gh,g^d=fgh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17720⋊C3, #17782⋊C3, #33373⋊C3, … |
| maximal normal subgroup of | #17720⋊C3, #17782⋊C3, #33373⋊C3, … |
| maximal quotient of | #34183⋊C3, #34258⋊C3, #34313⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18186);