Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini, socle | C2 | C23×S4 |
| Fitting | #585 | D3 |
| hypercenter | #41 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ #41 | |
738 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,16,17)(2,9,24,8,21,23)(3,27,28)(4,15,6)(7,19,32,18,29,31)(10,22)(11,26,13)(20,30);(1,3,6,13)(2,7,8,18)(4,11,16,27)(5,12,14,25)(9,19,21,29)(10,20,22,30)(15,26,17,28)(23,31,24,32);(1,2)(3,7)(4,9)(5,10)(6,8)(11,19)(12,20)(13,18)(14,22)(15,23)(16,21)(17,24)(25,30)(26,31)(27,29)(28,32);(4,14)(5,16)(9,22)(10,21)(11,25)(12,27)(15,17)(19,30)(20,29)(23,24)(26,28)(31,32)Polycyclic presentation:
2^4,3,2^3|c^2=h,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,d^b=dh,f^e=g,g^e=fgh,g^f=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #11712952, #11712954, #11714661, … |
| maximal normal subgroup of | #11712952, #11712954, #11714661, … |
| maximal quotient of | #11714606, #11714636, #11714641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20073);