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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e \mid a^{4}=b^{2}=c^{2}=d^{4}=e^{4}=[a,b]=[a,d]=[b,c]=[b,e]=[c,d]=[d,e]=1, c^{a}=cd^{2}e^{2}, e^{a}=ce^{3}, d^{b}=d^{3}, e^{c}=d^{2}e \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P
$(D_4\times C_2^3):C_4$ 1A $1$ $1$ $(D_4\times C_2^3):C_4$ 1A $1$
$(D_4\times C_2^3):C_4$ 2A $2$ $1$ $(D_4\times C_2^3):C_4$ 1A $d^{2}$
$(D_4\times C_2^3):C_4$ 2B $2$ $2$ $C_2^5:C_4$ 1A $b$
$(D_4\times C_2^3):C_4$ 2C $2$ $2$ $C_2^5:C_4$ 1A $be^{2}$
$(D_4\times C_2^3):C_4$ 2D $2$ $2$ $C_2^5:C_4$ 1A $bd$
$(D_4\times C_2^3):C_4$ 2E $2$ $2$ $C_2^5:C_4$ 1A $bde^{2}$
$(D_4\times C_2^3):C_4$ 2F $2$ $2$ $D_4^2:C_2$ 1A $e^{2}$
$(D_4\times C_2^3):C_4$ 2G $2$ $4$ $D_4\times C_2^3$ 1A $c$
$(D_4\times C_2^3):C_4$ 2H $2$ $4$ $C_2^3:D_4$ 1A $bc$
$(D_4\times C_2^3):C_4$ 2I $2$ $4$ $D_4\times C_2^3$ 1A $a^{2}e$
$(D_4\times C_2^3):C_4$ 2J $2$ $4$ $C_2^4:C_4$ 1A $a^{2}b$
$(D_4\times C_2^3):C_4$ 2K $2$ $4$ $C_2^3:D_4$ 1A $bcd$
$(D_4\times C_2^3):C_4$ 2L $2$ $4$ $D_4\times C_2^3$ 1A $a^{2}d^{2}e$
$(D_4\times C_2^3):C_4$ 2M $2$ $4$ $\OD_{16}:C_2^2$ 1A $a^{2}cd$
$(D_4\times C_2^3):C_4$ 2N $2$ $4$ $C_2^4:C_4$ 1A $a^{2}bd$
$(D_4\times C_2^3):C_4$ 2O $2$ $4$ $C_4^2:C_2^2$ 1A $a^{2}$
$(D_4\times C_2^3):C_4$ 2P $2$ $8$ $C_2^5$ 1A $a^{2}be$
$(D_4\times C_2^3):C_4$ 2Q $2$ $8$ $C_2^5$ 1A $a^{2}bde$
$(D_4\times C_2^3):C_4$ 4A $4$ $2$ $(C_2^3\times C_4):C_4$ 2A $d$
$(D_4\times C_2^3):C_4$ 4B $4$ $2$ $(C_2^3\times C_4):C_4$ 2A $de^{2}$
$(D_4\times C_2^3):C_4$ 4C $4$ $4$ $C_4^2:C_2^2$ 2A $cd$
$(D_4\times C_2^3):C_4$ 4D $4$ $4$ $C_2^3.C_2^3$ 2A $a^{2}d$
$(D_4\times C_2^3):C_4$ 4E $4$ $4$ $\OD_{16}.C_2^2$ 2A $a^{2}bc$
$(D_4\times C_2^3):C_4$ 4F $4$ $4$ $\OD_{16}.C_2^2$ 2A $a^{2}bcd$
$(D_4\times C_2^3):C_4$ 4G $4$ $4$ $C_8.C_2^3$ 2A $a^{2}c$
$(D_4\times C_2^3):C_4$ 4H $4$ $8$ $C_4\times D_4$ 2F $e$
$(D_4\times C_2^3):C_4$ 4I $4$ $8$ $C_4\times D_4$ 2F $de$
$(D_4\times C_2^3):C_4$ 4J $4$ $8$ $C_4\times D_4$ 2F $be$
$(D_4\times C_2^3):C_4$ 4K $4$ $8$ $C_4\times D_4$ 2F $bde$
$(D_4\times C_2^3):C_4$ 4L $4$ $8$ $C_2^3\times C_4$ 2A $a^{2}de$
$(D_4\times C_2^3):C_4$ 4M1 $4$ $8$ $C_4\times D_4$ 2O $a$
$(D_4\times C_2^3):C_4$ 4M-1 $4$ $8$ $C_4\times D_4$ 2O $a^{3}$
$(D_4\times C_2^3):C_4$ 4N1 $4$ $8$ $C_4\times D_4$ 2O $ad$
$(D_4\times C_2^3):C_4$ 4N-1 $4$ $8$ $C_4\times D_4$ 2O $a^{3}d$
$(D_4\times C_2^3):C_4$ 4O1 $4$ $8$ $C_4\times D_4$ 2O $ab$
$(D_4\times C_2^3):C_4$ 4O-1 $4$ $8$ $C_4\times D_4$ 2O $a^{3}b$
$(D_4\times C_2^3):C_4$ 4P1 $4$ $8$ $C_4\times D_4$ 2O $abd$
$(D_4\times C_2^3):C_4$ 4P-1 $4$ $8$ $C_4\times D_4$ 2O $a^{3}bd$
$(D_4\times C_2^3):C_4$ 8A1 $8$ $8$ $\OD_{16}:C_2$ 4G $ae$
$(D_4\times C_2^3):C_4$ 8A-1 $8$ $8$ $\OD_{16}:C_2$ 4G $a^{3}e$
$(D_4\times C_2^3):C_4$ 8B1 $8$ $8$ $\OD_{16}:C_2$ 4G $ade$
$(D_4\times C_2^3):C_4$ 8B-1 $8$ $8$ $\OD_{16}:C_2$ 4G $a^{3}de$
$(D_4\times C_2^3):C_4$ 8C1 $8$ $8$ $\OD_{16}:C_2$ 4G $abe$
$(D_4\times C_2^3):C_4$ 8C-1 $8$ $8$ $\OD_{16}:C_2$ 4G $a^{3}be$
$(D_4\times C_2^3):C_4$ 8D1 $8$ $8$ $\OD_{16}:C_2$ 4G $abde$
$(D_4\times C_2^3):C_4$ 8D-1 $8$ $8$ $\OD_{16}:C_2$ 4G $a^{3}bde$
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