Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3487⋊C3 |
| Fitting, Sylow | #3487 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3487 | C3 |
| hypercenter | #123 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #123 | |
366 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(6,12)(8,9,18)(11,22)(13,26,27,25,14,28)(15,16,29)(19,21,20)(23,34,35,33,24,36)(30,32,31);(2,4)(5,29)(6,35)(7,20)(8,32)(9,15)(10,31)(11,28)(12,36)(13,34)(14,23)(16,19)(17,18)(21,30)(22,27)(24,25)(26,33);(1,2)(3,4)(5,6,10,12)(7,11,17,22)(8,13,19,25)(9,14,21,26)(15,23,30,33)(16,24,32,34)(18,27,20,28)(29,35,31,36)Polycyclic presentation:
2^3,3,2^4|a^2=h,b^2=h,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=g,g^d=fg,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #33351⋊C3, #11710351, C2×#3487⋊C3, … |
| maximal normal subgroup of | #33351⋊C3, #11710351, C2×#3487⋊C3, … |
| maximal quotient of | #34174⋊C3, #34249⋊C3, #34257⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18182);