Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #504 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5402 |
| commutator, cosocle | #131⋊C3 | C22 |
| Frattini | C8 | C2×S4 |
| Fitting | #504 | D3 |
| hypercenter | C16 | S4 |
| hypocenter | SL(2,𝔽3) | D8 |
| LCS | G ▷ #131⋊C3 ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,37,58,25,41,62,17,26,21,13,29,63,14,33,47,8,22,39,9,59,50,10,15,55,7,11,57,32,18,61,3,38,23,2,46,64,4,51,44,28,5,36,31,53,49,24,45,56)(6,12,30,54,19,34,48,35,20,40,43,60,42,52,16,27);(2,8)(3,9)(4,14)(5,20)(6,22)(10,32)(11,35)(12,38)(13,28)(15,43)(16,46)(17,31)(18,19)(21,23)(24,25)(26,40)(27,37)(29,48)(30,45)(33,54)(34,53)(36,56)(39,55)(41,42)(44,58)(47,57)(51,60)(52,59)(61,63)(62,64)Polycyclic presentation:
2^4,3,2^3|b^2=cd,c^2=dh,d^2=h,f^2=h,g^2=h,b^a=bc,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #11711666, #11711667, #11711678, … |
| maximal normal subgroup of | #11711666, #11711667, #11711678, … |
| maximal quotient of | #11711470, #11711474, #11711477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5616);