Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #568 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×C4⋊S4 |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #568 | D3 |
| prime radical | #5923 | C2 |
| hypercenter | M16 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
243 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,51,42,18,6,50,3,47,29,4,15,61,8,19,44,14,27,53,5,39,26,21,23,62)(2,56,58,17,11,59,22,45,38,10,43,64,13,34,57,33,28,60,9,52,37,36,31,63)(7,20,24,48,30,54,16,40)(12,35,32,46,25,55,41,49);(1,3,8,5)(2,9,13,22)(4,18,21,14)(6,15,27,23)(7,16,30,24)(10,33,36,17)(11,31,28,43)(12,32,25,41)(19,47,51,39)(20,48,54,40)(26,42,29,44)(34,52,56,45)(35,49,55,46)(37,57,38,58)(50,62,53,61)(59,64,60,63);(1,2)(3,9)(4,17)(5,22)(6,25)(7,28)(8,13)(10,18)(11,30)(12,27)(14,36)(15,41)(16,43)(19,49)(20,52)(21,33)(23,32)(24,31)(26,38)(29,37)(34,40)(35,39)(42,58)(44,57)(45,54)(46,51)(47,55)(48,56)(50,64)(53,63)(59,61)(60,62)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=d,d^2=h,f^2=h,g^2=h,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11713371, #11713373, #11713431, … |
| maximal normal subgroup of | #11713371, #11713373, #11713431, … |
| maximal quotient of | #11712178, #11712179, #11712182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18031);