Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2222 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C11×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C11×#131 ◁ G | |
140 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(12,13,19,25)(14,26,32,54)(15,33,36,50)(16,37,39,48)(17,24,18,23)(20,41,45,29)(21,40,47,35)(22,38,49,27)(28,57,60,74)(30,53,31,52)(34,65,66,46)(42,69,70,75)(43,61,44,55)(51,62,73,59)(56,64,67,72)(58,63,68,71);(12,14,27,55,18,31,39,41,19,32,38,61,17,30,16,29)(13,20,37,53,24,44,49,54,25,45,48,52,23,43,22,26)(15,28,56,75,35,59,66,68,36,60,67,69,40,62,34,58)(21,42,64,57,33,63,65,73,47,70,72,74,50,71,46,51);(12,15)(13,21)(14,28)(16,34)(17,40)(18,35)(19,36)(20,42)(22,46)(23,50)(24,33)(25,47)(26,51)(27,56)(29,58)(30,62)(31,59)(32,60)(37,64)(38,67)(39,66)(41,68)(43,71)(44,63)(45,70)(48,72)(49,65)(52,74)(53,57)(54,73)(55,75)(61,69)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 954);