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Elements 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
$C_5:F_5$	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	$bc^{2}$

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