Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2262 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | D16 |
| commutator, Frattini, cosocle | C16 | C23×C11 |
| socle | C2×C11 | C2×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ C2×C11×C16 ◁ G | |
106 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)(14,25)(15,20)(16,34)(17,38)(18,24)(19,27)(21,36)(22,32)(23,33)(26,52)(28,58)(29,60)(30,53)(31,61)(35,49)(37,67)(39,46)(40,45)(41,54)(42,68)(43,57)(44,63)(47,73)(48,70)(50,65)(51,56)(55,72)(59,74)(62,71)(64,66)(69,75);(12,14)(13,19)(15,26)(16,35)(17,39)(18,30)(20,41)(21,46)(22,49)(23,51)(24,43)(25,34)(27,36)(28,32)(29,33)(31,62)(37,69)(38,42)(40,55)(44,58)(45,71)(47,74)(48,72)(50,75)(52,61)(53,67)(54,63)(56,73)(57,70)(59,65)(60,66)(64,68);(12,15,18,33)(13,20,24,23)(14,26,30,29)(16,31,37,64)(17,32,40,65)(19,41,43,51)(21,44,48,47)(22,45,50,38)(25,52,53,60)(27,54,57,56)(28,55,59,39)(34,61,67,66)(35,62,69,68)(36,63,70,73)(42,49,71,75)(46,58,72,74)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 994);