Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3428⋊C3 |
| Fitting, Sylow | #3428 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | #3428 | C3 |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | C2×Q8 | C23×C3 |
| hypercenter | #117 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
365 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,7,8,15)(6,12,11,22)(9,17,21,29,10,31,18,30,20,16,19,32)(13,26,28,33,14,35,23,34,27,25,24,36);(5,20,8,21)(6,28,11,27)(7,31,15,32)(9,19,18,10)(12,36,22,35)(13,14,23,24)(16,30,29,17)(25,26,33,34);(1,2,3,4)(5,6)(7,12)(8,11)(9,13)(10,14)(15,22)(16,25)(17,26)(18,23)(19,24)(20,27)(21,28)(29,33)(30,34)(31,35)(32,36)Polycyclic presentation:
2^3,3,2^4|a^2=eh,c^2=h,f^2=h,g^2=h,b^a=bh,f^d=g,g^d=fg,g^f=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #20338⋊C3, #20343⋊C3, #20360⋊C3, … |
| maximal normal subgroup of | #20338⋊C3, #20343⋊C3, #20360⋊C3, … |
| maximal quotient of | #11709754, C4×#520⋊C3, C4×#519⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18155);