Finite Group DatabasebetaFinite Group Databasebeta| order | 3136 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C72 · Schur multiplier (H2) = C23×C7 |
| direct product | C72×#442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C72 | C2×D4 |
| commutator | C4 | C22×C4×C72 |
| Frattini, cosocle | C2×C4 | C23×C72 |
| prime radical | C2×C72×D8 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C72 ◁ C42×C72 ◁ G | |
660 conjugacy classes of subgroups; only the 350 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 350 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1568 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,4,3)(6,9)(7,11)(10,12)(13,14,15,16,17,18,19);(1,3,4,2)(6,10)(9,12)(20,21,22,23,24,25,26);(5,6,7,9,8,10,11,12)1078 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | #442×C7⋊D7, #442×#3861 |
| maximal subgroup of | #442×C7⋊D7, #442×#3861 |
| maximal normal subgroup of | #442×C7⋊D7, #442×#3861 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,3)(6,9)(7,11)(10,12)(13,14,15,16,17,18,19), (1,3,4,2)(6,10)(9,12)(20,21,22,23,24,25,26), (5,6,7,9,8,10,11,12));