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Download displayed columns for resultsElements of the group are displayed as words in the presentation $\langle a, b, c \mid a^{8}=b^{3}=c^{3}=[b,c]=1, b^{a}=bc, c^{a}=b^{2}c \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $F_9$ | 1A | $1$ | $1$ | $F_9$ | 1A | 1A | $1$ |
| $F_9$ | 2A | $2$ | $9$ | $C_8$ | 1A | 2A | $a^{4}c^{2}$ |
| $F_9$ | 3A | $3$ | $8$ | $C_3^2$ | 3A | 1A | $c$ |
| $F_9$ | 4A1 | $4$ | $9$ | $C_8$ | 2A | 4A-1 | $a^{2}b^{2}c$ |
| $F_9$ | 4A-1 | $4$ | $9$ | $C_8$ | 2A | 4A1 | $a^{6}bc$ |
| $F_9$ | 8A1 | $8$ | $9$ | $C_8$ | 4A1 | 8A3 | $a^{5}b^{2}c^{2}$ |
| $F_9$ | 8A-1 | $8$ | $9$ | $C_8$ | 4A-1 | 8A-3 | $a^{3}b^{2}$ |
| $F_9$ | 8A3 | $8$ | $9$ | $C_8$ | 4A-1 | 8A1 | $a^{7}bc^{2}$ |
| $F_9$ | 8A-3 | $8$ | $9$ | $C_8$ | 4A1 | 8A-1 | $ab$ |