Group information
| Description: | $\GOrthMinus(6,7)$ | |
| Order: | \(9324577382400\)\(\medspace = 2^{13} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 43 \) | |
| Exponent: | \(361200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 43 \) | |
| Automorphism group: | Group of order \(37298309529600\)\(\medspace = 2^{15} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 43 \) | |
| Composition factors: | $C_2$ x 3, $\PSU(4,7)$ | |
| Derived length: | $1$ |
This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.
Group statistics
| Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 10 | 12 | 14 | 15 | 16 | 21 | 24 | 25 | 28 | 30 | 42 | 43 | 48 | 50 | 56 | 75 | 84 | 86 | 150 | 168 | 172 | 344 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 53100615 | 295687952 | 18294865400 | 15540962304 | 209024951856 | 13841287200 | 330584497600 | 233114434560 | 438245886400 | 335544789600 | 31081924608 | 388524057600 | 29733984000 | 1022188294400 | 77704811520 | 319569532800 | 93245773824 | 755243193600 | 189744307200 | 777048115200 | 1165572172800 | 764021798400 | 155409623040 | 55503436800 | 189744307200 | 466228869120 | 111006873600 | 379488614400 | 758977228800 | 9324577382400 |
| Conjugacy classes | 1 | 11 | 2 | 13 | 1 | 26 | 6 | 45 | 7 | 16 | 38 | 1 | 14 | 3 | 38 | 5 | 16 | 3 | 21 | 7 | 28 | 35 | 38 | 5 | 2 | 7 | 15 | 4 | 14 | 28 | 450 |
| Divisions | 1 | 11 | 2 | 13 | 1 | 26 | 6 | 25 | 7 | 16 | 38 | 1 | 7 | 3 | 19 | 1 | 16 | 3 | 21 | 1 | 7 | 7 | 19 | 1 | 2 | 1 | 3 | 2 | 1 | 1 | 262 |
| Autjugacy classes | 1 | 6 | 2 | 9 | 1 | 13 | 4 | 24 | 3 | 11 | 15 | 1 | 10 | 2 | 22 | 5 | 10 | 2 | 8 | 7 | 16 | 15 | 17 | 5 | 2 | 7 | 10 | 2 | 14 | 14 | 258 |
Minimal presentations
| Permutation degree: | not computed |
| Transitive degree: | not computed |
| Rank: | $2$ |
| Inequivalent generating pairs: | not computed |
Minimal degrees of faithful linear representations
| Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
|---|---|---|---|
| Irreducible | 600 | not computed | not computed |
| Arbitrary | not computed | not computed | not computed |
Constructions
| Groups of Lie type: | $\GOMinus(6,7)$ | ||||
| Direct product: | not computed | ||||
| Semidirect product: | not computed | ||||
| Trans. wreath product: | not computed | ||||
Elements of the group are displayed as matrices in $\GOMinus(6,7)$.
Homology
| Abelianization: | $C_{2}^{2} $ | |
| Schur multiplier: | not computed | |
| Commutator length: | $1$ |
Subgroups
Subgroup data has not been computed.
Character theory
Complex character table
The $450 \times 450$ character table is not available for this group.
Rational character table
The $262 \times 262$ rational character table is not available for this group.