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Elements of the group are displayed as words in the presentation $\langle a \mid a^{1002}=1 \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative | ||
|---|---|---|---|---|---|---|---|---|
| 2P | 3P | 167P | ||||||
| $C_{1002}$ | 1A | $1$ | $1$ | $C_{1002}$ | 1A | 1A | 1A | $1$ |
| $C_{1002}$ | 2A | $2$ | $1$ | $C_{1002}$ | 1A | 2A | 2A | $a^{501}$ |
| $C_{1002}$ | 3A1 | $3$ | $1$ | $C_{1002}$ | 3A-1 | 1A | 3A-1 | $a^{334}$ |
| $C_{1002}$ | 3A-1 | $3$ | $1$ | $C_{1002}$ | 3A1 | 1A | 3A1 | $a^{668}$ |
| $C_{1002}$ | 6A1 | $6$ | $1$ | $C_{1002}$ | 3A1 | 2A | 6A-1 | $a^{167}$ |
| $C_{1002}$ | 6A-1 | $6$ | $1$ | $C_{1002}$ | 3A-1 | 2A | 6A1 | $a^{835}$ |
| $C_{1002}$ | 167A1 | $167$ | $1$ | $C_{1002}$ | 167A2 | 167A3 | 167A83 | $a^{6}$ |
| $C_{1002}$ | 167A-1 | $167$ | $1$ | $C_{1002}$ | 167A-2 | 167A-3 | 167A-83 | $a^{996}$ |
| $C_{1002}$ | 167A2 | $167$ | $1$ | $C_{1002}$ | 167A4 | 167A6 | 167A-1 | $a^{12}$ |
| $C_{1002}$ | 167A-2 | $167$ | $1$ | $C_{1002}$ | 167A-4 | 167A-6 | 167A1 | $a^{990}$ |
| $C_{1002}$ | 167A3 | $167$ | $1$ | $C_{1002}$ | 167A6 | 167A9 | 167A82 | $a^{18}$ |
| $C_{1002}$ | 167A-3 | $167$ | $1$ | $C_{1002}$ | 167A-6 | 167A-9 | 167A-82 | $a^{984}$ |
| $C_{1002}$ | 167A4 | $167$ | $1$ | $C_{1002}$ | 167A8 | 167A12 | 167A-2 | $a^{24}$ |
| $C_{1002}$ | 167A-4 | $167$ | $1$ | $C_{1002}$ | 167A-8 | 167A-12 | 167A2 | $a^{978}$ |
| $C_{1002}$ | 167A5 | $167$ | $1$ | $C_{1002}$ | 167A10 | 167A15 | 167A81 | $a^{30}$ |
| $C_{1002}$ | 167A-5 | $167$ | $1$ | $C_{1002}$ | 167A-10 | 167A-15 | 167A-81 | $a^{972}$ |
| $C_{1002}$ | 167A6 | $167$ | $1$ | $C_{1002}$ | 167A12 | 167A18 | 167A-3 | $a^{36}$ |
| $C_{1002}$ | 167A-6 | $167$ | $1$ | $C_{1002}$ | 167A-12 | 167A-18 | 167A3 | $a^{966}$ |
| $C_{1002}$ | 167A7 | $167$ | $1$ | $C_{1002}$ | 167A14 | 167A21 | 167A80 | $a^{42}$ |
| $C_{1002}$ | 167A-7 | $167$ | $1$ | $C_{1002}$ | 167A-14 | 167A-21 | 167A-80 | $a^{960}$ |
| $C_{1002}$ | 167A8 | $167$ | $1$ | $C_{1002}$ | 167A16 | 167A24 | 167A-4 | $a^{48}$ |
| $C_{1002}$ | 167A-8 | $167$ | $1$ | $C_{1002}$ | 167A-16 | 167A-24 | 167A4 | $a^{954}$ |
| $C_{1002}$ | 167A9 | $167$ | $1$ | $C_{1002}$ | 167A18 | 167A27 | 167A79 | $a^{54}$ |
| $C_{1002}$ | 167A-9 | $167$ | $1$ | $C_{1002}$ | 167A-18 | 167A-27 | 167A-79 | $a^{948}$ |
| $C_{1002}$ | 167A10 | $167$ | $1$ | $C_{1002}$ | 167A20 | 167A30 | 167A-5 | $a^{60}$ |
| $C_{1002}$ | 167A-10 | $167$ | $1$ | $C_{1002}$ | 167A-20 | 167A-30 | 167A5 | $a^{942}$ |
| $C_{1002}$ | 167A11 | $167$ | $1$ | $C_{1002}$ | 167A22 | 167A33 | 167A78 | $a^{66}$ |
| $C_{1002}$ | 167A-11 | $167$ | $1$ | $C_{1002}$ | 167A-22 | 167A-33 | 167A-78 | $a^{936}$ |
| $C_{1002}$ | 167A12 | $167$ | $1$ | $C_{1002}$ | 167A24 | 167A36 | 167A-6 | $a^{72}$ |
| $C_{1002}$ | 167A-12 | $167$ | $1$ | $C_{1002}$ | 167A-24 | 167A-36 | 167A6 | $a^{930}$ |
| $C_{1002}$ | 167A13 | $167$ | $1$ | $C_{1002}$ | 167A26 | 167A39 | 167A77 | $a^{78}$ |
| $C_{1002}$ | 167A-13 | $167$ | $1$ | $C_{1002}$ | 167A-26 | 167A-39 | 167A-77 | $a^{924}$ |
| $C_{1002}$ | 167A14 | $167$ | $1$ | $C_{1002}$ | 167A28 | 167A42 | 167A-7 | $a^{84}$ |
| $C_{1002}$ | 167A-14 | $167$ | $1$ | $C_{1002}$ | 167A-28 | 167A-42 | 167A7 | $a^{918}$ |
| $C_{1002}$ | 167A15 | $167$ | $1$ | $C_{1002}$ | 167A30 | 167A45 | 167A76 | $a^{90}$ |
| $C_{1002}$ | 167A-15 | $167$ | $1$ | $C_{1002}$ | 167A-30 | 167A-45 | 167A-76 | $a^{912}$ |
| $C_{1002}$ | 167A16 | $167$ | $1$ | $C_{1002}$ | 167A32 | 167A48 | 167A-8 | $a^{96}$ |
| $C_{1002}$ | 167A-16 | $167$ | $1$ | $C_{1002}$ | 167A-32 | 167A-48 | 167A8 | $a^{906}$ |
| $C_{1002}$ | 167A17 | $167$ | $1$ | $C_{1002}$ | 167A34 | 167A51 | 167A75 | $a^{102}$ |
| $C_{1002}$ | 167A-17 | $167$ | $1$ | $C_{1002}$ | 167A-34 | 167A-51 | 167A-75 | $a^{900}$ |
| $C_{1002}$ | 167A18 | $167$ | $1$ | $C_{1002}$ | 167A36 | 167A54 | 167A-9 | $a^{108}$ |
| $C_{1002}$ | 167A-18 | $167$ | $1$ | $C_{1002}$ | 167A-36 | 167A-54 | 167A9 | $a^{894}$ |
| $C_{1002}$ | 167A19 | $167$ | $1$ | $C_{1002}$ | 167A38 | 167A57 | 167A74 | $a^{114}$ |
| $C_{1002}$ | 167A-19 | $167$ | $1$ | $C_{1002}$ | 167A-38 | 167A-57 | 167A-74 | $a^{888}$ |
| $C_{1002}$ | 167A20 | $167$ | $1$ | $C_{1002}$ | 167A40 | 167A60 | 167A-10 | $a^{120}$ |
| $C_{1002}$ | 167A-20 | $167$ | $1$ | $C_{1002}$ | 167A-40 | 167A-60 | 167A10 | $a^{882}$ |
| $C_{1002}$ | 167A21 | $167$ | $1$ | $C_{1002}$ | 167A42 | 167A63 | 167A73 | $a^{126}$ |
| $C_{1002}$ | 167A-21 | $167$ | $1$ | $C_{1002}$ | 167A-42 | 167A-63 | 167A-73 | $a^{876}$ |
| $C_{1002}$ | 167A22 | $167$ | $1$ | $C_{1002}$ | 167A44 | 167A66 | 167A-11 | $a^{132}$ |
| $C_{1002}$ | 167A-22 | $167$ | $1$ | $C_{1002}$ | 167A-44 | 167A-66 | 167A11 | $a^{870}$ |