Finite Group DatabasebetaFinite Group Databasebeta| order | 1824 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C19 · Schur multiplier (H2) = C22 |
| direct product | C3×C19×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C19 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C3×C19 |
| socle | C2×C3×C19 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C19 ◁ C2×C3×C4×C19 ◁ G | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 912 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24)(25,32)(26,30)(27,35)(28,31)(29,33)(34,37)(36,38);(23,25)(24,29)(26,33)(27,36)(28,34)(30,37)(31,38)(32,35);(23,26,28,27)(24,30,31,35)(25,33,34,36)(29,37,38,32)798 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1824, 1040);