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Download displayed columns for resultsElements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{8}=c^{2}=d^{2}=[a,d]=[b,d]=[c,d]=1, b^{a}=b^{3}, c^{a}=b^{4}c, c^{b}=cd \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative |
|---|---|---|---|---|---|---|
| 2P | ||||||
| $D_4.D_4$ | 1A | $1$ | $1$ | $D_4.D_4$ | 1A | $1$ |
| $D_4.D_4$ | 2A | $2$ | $1$ | $D_4.D_4$ | 1A | $b^{4}d$ |
| $D_4.D_4$ | 2B | $2$ | $1$ | $D_4.D_4$ | 1A | $b^{4}$ |
| $D_4.D_4$ | 2C | $2$ | $1$ | $D_4.D_4$ | 1A | $d$ |
| $D_4.D_4$ | 2D | $2$ | $4$ | $C_2^2\times C_4$ | 1A | $c$ |
| $D_4.D_4$ | 2E | $2$ | $4$ | $C_2^2\times C_4$ | 1A | $a$ |
| $D_4.D_4$ | 2F | $2$ | $4$ | $C_2^2\times C_4$ | 1A | $ad$ |
| $D_4.D_4$ | 4A | $4$ | $2$ | $C_2^2:C_8$ | 2B | $b^{2}$ |
| $D_4.D_4$ | 4B | $4$ | $2$ | $C_2^2:C_8$ | 2B | $b^{2}d$ |
| $D_4.D_4$ | 4C1 | $4$ | $2$ | $D_4:C_2^2$ | 2B | $b^{2}c$ |
| $D_4.D_4$ | 4C-1 | $4$ | $2$ | $D_4:C_2^2$ | 2B | $b^{6}c$ |
| $D_4.D_4$ | 4D | $4$ | $4$ | $C_2^2\times C_4$ | 2B | $ac$ |
| $D_4.D_4$ | 4E | $4$ | $4$ | $C_2^2\times C_4$ | 2B | $acd$ |
| $D_4.D_4$ | 4F | $4$ | $8$ | $C_2\times C_4$ | 2B | $ab$ |
| $D_4.D_4$ | 4G | $4$ | $8$ | $C_2\times C_4$ | 2C | $abc$ |
| $D_4.D_4$ | 8A1 | $8$ | $4$ | $C_2\times C_8$ | 4A | $b$ |
| $D_4.D_4$ | 8A-1 | $8$ | $4$ | $C_2\times C_8$ | 4A | $b^{5}$ |
| $D_4.D_4$ | 8B1 | $8$ | $4$ | $C_2\times C_8$ | 4B | $bc$ |
| $D_4.D_4$ | 8B3 | $8$ | $4$ | $C_2\times C_8$ | 4B | $b^{5}c$ |