Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#3417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C11 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C11×#41 ◁ C11×#131 ◁ G | |
350 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,18,23)(14,24,28,52)(15,20,29,43)(16,32,36,50)(17,37,34,22)(19,39,42,26)(21,46,48,35)(25,51,53,73)(27,56,38,47)(30,59,57,72)(31,62,58,45)(33,49,64,41)(40,65,66,54)(44,69,68,61)(55,74,63,70)(60,71,75,67);(12,14,18,28)(13,19,23,42)(15,25,29,53)(16,33,36,64)(17,38,34,27)(20,40,43,66)(21,47,48,56)(22,49,37,41)(24,50,52,32)(26,46,39,35)(30,60,57,75)(31,63,58,55)(44,70,68,74)(45,71,62,67)(51,72,73,59)(54,69,65,61);(12,15)(13,20)(14,25)(16,30)(17,31)(18,29)(19,40)(21,44)(22,45)(23,43)(24,51)(26,54)(27,55)(28,53)(32,59)(33,60)(34,58)(35,61)(36,57)(37,62)(38,63)(39,65)(41,67)(42,66)(46,69)(47,70)(48,68)(49,71)(50,72)(52,73)(56,74)(64,75);(15,29)(20,43)(25,53)(30,57)(31,58)(40,66)(44,68)(45,62)(51,73)(54,65)(55,63)(59,72)(60,75)(61,69)(67,71)(70,74)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2149);