Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1417 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | D16 |
| commutator | C16 | C2×C3×C4 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| socle | C2×C3 | #357 |
| prime radical | C3×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,5,10,7,13,26,12)(3,14,16,43,19,44,18,50)(4,20,23,51,25,53,58,52)(6,27,22,11,29,39,57,41)(8,30,31,46,33,45,42,15)(9,34,36,59,38,61,37,24)(17,48,47,62,49,63,28,35)(21,40,54,60,56,64,55,32)(65,66,67);(1,3)(2,8)(4,21)(5,16)(6,28)(7,19)(9,35)(10,31)(11,40)(12,42)(13,33)(14,20)(15,24)(17,22)(18,26)(23,54)(25,56)(27,32)(29,47)(30,34)(36,48)(37,63)(38,62)(39,60)(41,64)(43,51)(44,53)(45,61)(46,59)(49,57)(50,52)(55,58)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 553);