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×#2192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C11×#131 ◁ G | |
220 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 220 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) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,25)(15,29)(16,23)(17,22)(18,24)(19,26)(20,27)(21,32)(28,40)(30,41)(31,35)(33,39)(34,38)(36,43)(37,42);(12,14)(13,19)(15,30)(16,27)(17,34)(18,28)(20,35)(21,38)(22,36)(23,41)(24,37)(25,29)(26,32)(31,42)(33,43)(39,40);(12,14)(13,20)(15,26)(16,34)(17,27)(18,28)(19,35)(21,37)(22,40)(23,25)(24,38)(29,41)(30,32)(31,43)(33,42)(36,39)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 924);