Finite Group DatabasebetaFinite Group Databasebeta| order | 2624 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C41 · Schur multiplier (H2) = C26 |
| direct product | C2×C4×D4×C41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C41 | C22 |
| commutator | C2 | C23×C4×C41 |
| Frattini, cosocle | C22 | C24×C41 |
| socle | C23×C41 | C23 |
| prime radical | C22×D4×C41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C41 ◁ G | |
426 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1312 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(3,4,5,6);(7,8,9,10);(8,10)1640 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C41×#1459, C41×#1460, C41×#1465, … |
| maximal normal subgroup of | C41×#1459, C41×#1460, C41×#1465, … |
| maximal quotient of | C2×C4×C22⋊C4×C41, C2×C4×C4⋊C4×C41, C42×D4×C41, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2624, 196);