Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2195 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C11×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C11×#131 ◁ G | |
206 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 64 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)(14,26,30,54)(15,31,34,51)(16,35,39,49)(17,24,18,23)(20,42,45,28)(21,40,47,33)(22,37,50,38)(27,55,59,72)(29,60,41,48)(32,63,66,46)(36,52,67,44)(43,64,71,75)(53,58,74,57)(56,61,69,70)(62,68,73,65);(12,14)(13,20)(15,27)(16,36)(17,29)(18,41)(19,30)(21,43)(22,48)(23,44)(24,52)(25,45)(26,35)(28,38)(31,61)(32,64)(33,57)(34,59)(37,42)(39,67)(40,58)(46,72)(47,71)(49,54)(50,60)(51,70)(53,62)(55,63)(56,65)(66,75)(68,69)(73,74);(12,15,19,34)(13,21,25,47)(14,27,30,59)(16,32,39,66)(17,40,18,33)(20,43,45,71)(22,46,50,63)(23,51,24,31)(26,53,54,74)(28,56,42,69)(29,58,41,57)(35,62,49,73)(36,64,67,75)(37,68,38,65)(44,70,52,61)(48,72,60,55)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 927);