Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #550 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | #550 | D3 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
276 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 34 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:
(2,4)(5,16,22,7,23,32,11,19,25,8,9,34)(6,33,29,18,24,36,15,27,30,12,13,35)(10,20,26,17)(14,28,21,31);(1,2)(3,4)(5,7,11,8)(6,12,15,18)(9,16,23,19)(10,17,26,20)(13,27,24,33)(14,28,21,31)(22,32,25,34)(29,35,30,36);(5,6)(7,12)(8,18)(9,21)(10,24)(11,15)(13,26)(14,23)(16,31)(17,33)(19,28)(20,27)(22,30)(25,29)(32,36)(34,35)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=h,f^2=h,g^2=h,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#85409, #11714881, #11714906, … |
| maximal normal subgroup of | C2×#85409, #11714881, #11714906, … |
| maximal quotient of | #11712270, #11712278, #11712288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17985);