Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2264 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×D16 |
| commutator, Frattini, cosocle | C16 | C23×C11 |
| prime radical | C11×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ C2×C11×C16 ◁ G | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,24)(15,29)(16,35)(17,39)(18,20)(19,25)(21,32)(22,31)(23,37)(26,56)(27,59)(28,51)(30,48)(33,67)(34,50)(36,46)(38,69)(40,52)(41,55)(42,63)(43,62)(44,53)(45,57)(47,74)(49,54)(58,75)(60,65)(61,66)(64,73)(68,71)(70,72);(12,14,18,28)(13,19,20,41)(15,30,34,64)(16,36,38,70)(17,40,37,27)(21,46,44,72)(22,48,45,73)(23,49,39,43)(24,50,51,29)(25,53,55,32)(26,57,58,31)(33,68,66,63)(35,42,69,71)(47,75,60,56)(52,61,59,67)(54,65,62,74);(13,20)(19,41)(21,44)(22,45)(23,39)(24,51)(29,50)(35,69)(42,71)(43,49)(46,72)(47,60)(48,73)(52,59)(56,75)(61,67)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 996);