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Download displayed columns for resultsElements of the group are displayed as words in the presentation $\langle a, b, c \mid a^{4}=b^{5}=c^{5}=[b,c]=1, b^{a}=b^{2}, c^{a}=c^{4} \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 5P | ||||||
| $C_5:F_5$ | 1A | $1$ | $1$ | $C_5:F_5$ | 1A | 1A | $1$ |
| $C_5:F_5$ | 2A | $2$ | $5$ | $C_5:C_4$ | 1A | 2A | $a^{2}$ |
| $C_5:F_5$ | 4A1 | $4$ | $25$ | $C_4$ | 2A | 4A1 | $ab^{4}c^{4}$ |
| $C_5:F_5$ | 4A-1 | $4$ | $25$ | $C_4$ | 2A | 4A-1 | $a^{3}b^{3}c^{4}$ |
| $C_5:F_5$ | 5A1 | $5$ | $2$ | $C_5\times D_5$ | 5A2 | 1A | $c^{2}$ |
| $C_5:F_5$ | 5A2 | $5$ | $2$ | $C_5\times D_5$ | 5A1 | 1A | $c^{4}$ |
| $C_5:F_5$ | 5B | $5$ | $4$ | $C_5^2$ | 5B | 1A | $b$ |
| $C_5:F_5$ | 5C1 | $5$ | $4$ | $C_5^2$ | 5C2 | 1A | $bc$ |
| $C_5:F_5$ | 5C-1 | $5$ | $4$ | $C_5^2$ | 5C-2 | 1A | $b^{2}c$ |
| $C_5:F_5$ | 5C2 | $5$ | $4$ | $C_5^2$ | 5C-1 | 1A | $bc^{3}$ |
| $C_5:F_5$ | 5C-2 | $5$ | $4$ | $C_5^2$ | 5C1 | 1A | $bc^{2}$ |
| $C_5:F_5$ | 10A1 | $10$ | $10$ | $C_{10}$ | 5A1 | 2A | $a^{2}c$ |
| $C_5:F_5$ | 10A3 | $10$ | $10$ | $C_{10}$ | 5A2 | 2A | $a^{2}c^{2}$ |