Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3488⋊C3 |
| Fitting, Sylow | #3488 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3488 | C3 |
| hypercenter | C22⋊Q8 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
383 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,15,25,10,8,23,11,17,16,7,18,26)(6,20,28,9,13,27,14,19,21,12,22,24);(2,4)(5,14,11,6)(7,9,10,12)(8,20,18,19)(13,15,22,17)(16,23,25,26)(21,24,28,27);(1,2)(3,4)(5,6,11,14)(7,9,10,12)(8,13,18,22)(15,19,17,20)(16,21,25,28)(23,24,26,27)Polycyclic presentation:
2^3,3,2^4|a^2=h,c^2=h,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=fg,g^d=f,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17611⋊C3, #60372⋊C3, #60374⋊C3, … |
| maximal normal subgroup of | #17611⋊C3, #60372⋊C3, #60374⋊C3, … |
| maximal quotient of | #34175⋊C3, #34250⋊C3, #34258⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18183);