Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f, g \mid b^{4}=c^{2}=d^{10}=e^{10}=f^{5}=g^{5}=[d,g]=[f,g]=1, a^{2}=b^{2}cd^{6}e^{8}f^{2}g^{2}, b^{a}=b^{3}ce^{6}fg^{2}, c^{a}=ce^{6}, d^{a}=b^{2}d^{5}e^{8}g, e^{a}=d^{6}e^{5}f^{4}g^{4}, f^{a}=g^{4}, g^{a}=f^{4}, c^{b}=cd^{2}e^{6}f^{2}g^{3}, d^{b}=d^{5}e^{2}fg^{3}, e^{b}=cd^{2}e^{3}f, f^{b}=e^{2}, g^{b}=d^{8}, d^{c}=d^{9}e^{4}f, e^{c}=e^{9}f^{3}, f^{c}=f^{4}, g^{c}=g^{4}, e^{d}=e^{9}f^{2}, f^{d}=f^{4}, f^{e}=f^{4}, g^{e}=g^{4} \rangle$ .
| Group |
Label |
Order |
Size |
Centralizer |
Powers |
Representative |
| 2P |
5P |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
1A |
$1$ |
$1$ |
$(\SO(3,7)\times S_4^2).C_2^2$ |
1A |
1A |
$1$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2A |
$2$ |
$1$ |
$(\SO(3,7)\times S_4^2).C_2^2$ |
1A |
2A |
$b^{2}cd^{8}f^{2}g^{2}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2B |
$2$ |
$50$ |
$C_{10}^2.D_4$ |
1A |
2B |
$e^{5}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2C |
$2$ |
$50$ |
$C_{10}^2.D_4$ |
1A |
2C |
$d^{5}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2D |
$2$ |
$50$ |
$C_{10}^2.D_4$ |
1A |
2D |
$cde^{2}f^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2E |
$2$ |
$50$ |
$C_{10}^2.D_4$ |
1A |
2E |
$b^{2}efg^{2}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2F |
$2$ |
$100$ |
$D_{10}^2$ |
1A |
2F |
$d^{5}ef$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2G |
$2$ |
$100$ |
$C_{10}^2.C_2^2$ |
1A |
2G |
$abd^{5}e^{6}f^{2}g^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2H |
$2$ |
$625$ |
$C_4^2:C_2^2$ |
1A |
2H |
$b^{2}d^{8}e^{4}f^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
2I |
$2$ |
$625$ |
$C_4^2:C_2^2$ |
1A |
2I |
$cfg^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4A1 |
$4$ |
$50$ |
$C_4\times D_5\wr C_2$ |
2A |
4A1 |
$bde^{3}f^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4A-1 |
$4$ |
$50$ |
$C_4\times D_5\wr C_2$ |
2A |
4A-1 |
$b^{3}cd^{9}e^{3}fg^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4B1 |
$4$ |
$50$ |
$C_4\times D_5\wr C_2$ |
2A |
4B1 |
$ad^{8}e^{9}f^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4B-1 |
$4$ |
$50$ |
$C_4\times D_5\wr C_2$ |
2A |
4B-1 |
$ab^{2}cd^{6}e^{9}f^{2}g^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4C |
$4$ |
$100$ |
$C_4\times D_5^2$ |
2A |
4C |
$ad^{6}e^{6}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4D |
$4$ |
$100$ |
$C_4\times D_5^2$ |
2A |
4D |
$be^{9}f^{4}g^{2}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4E |
$4$ |
$100$ |
$C_{20}:F_5$ |
2A |
4E |
$abcd^{2}e^{7}g^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4F1 |
$4$ |
$1250$ |
$C_4^2:C_2$ |
2I |
4F1 |
$acd^{5}ef^{2}g$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4F-1 |
$4$ |
$1250$ |
$C_4^2:C_2$ |
2I |
4F-1 |
$ad^{3}e^{5}f^{2}g^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4G1 |
$4$ |
$1250$ |
$C_4^2:C_2$ |
2H |
4G1 |
$bcd^{3}e^{8}f^{2}g$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4G-1 |
$4$ |
$1250$ |
$C_4^2:C_2$ |
2H |
4G-1 |
$b^{3}cdg^{2}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4H |
$4$ |
$2500$ |
$C_2^2\times C_4$ |
2H |
4H |
$bd^{4}e^{4}g^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4I |
$4$ |
$2500$ |
$C_4^2$ |
2I |
4I |
$ab^{3}cde^{7}f^{4}g$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4J |
$4$ |
$2500$ |
$C_4^2$ |
2H |
4J |
$abd^{6}e^{8}f^{3}g^{2}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
4K |
$4$ |
$2500$ |
$C_2^2\times C_4$ |
2I |
4K |
$acd^{9}e^{4}fg$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5A1 |
$5$ |
$8$ |
$C_{100}\times D_{25}$ |
5A2 |
1A |
$d^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5A2 |
$5$ |
$8$ |
$C_{100}\times D_{25}$ |
5A1 |
1A |
$d^{8}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5B1 |
$5$ |
$8$ |
$C_5\times C_{10}.D_5^2$ |
5B2 |
1A |
$d^{8}e^{2}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5B2 |
$5$ |
$8$ |
$C_5\times C_{10}.D_5^2$ |
5B1 |
1A |
$d^{6}e^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5C1 |
$5$ |
$8$ |
$C_5\times C_{10}.D_5^2$ |
5C2 |
1A |
$d^{2}g^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5C2 |
$5$ |
$8$ |
$C_5\times C_{10}.D_5^2$ |
5C1 |
1A |
$d^{4}g^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5D1 |
$5$ |
$8$ |
$C_5\times C_{10}.D_5^2$ |
5D2 |
1A |
$d^{2}e^{8}fg^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5D2 |
$5$ |
$8$ |
$C_5\times C_{10}.D_5^2$ |
5D1 |
1A |
$d^{4}e^{6}f^{2}g^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5E1 |
$5$ |
$8$ |
$C_5^4:Q_8$ |
5E2 |
1A |
$d^{2}e^{8}f^{4}g^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5E2 |
$5$ |
$8$ |
$C_5^4:Q_8$ |
5E1 |
1A |
$d^{4}e^{6}f^{3}g^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5F1 |
$5$ |
$8$ |
$C_5^3:D_{20}$ |
5F2 |
1A |
$d^{4}f^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5F2 |
$5$ |
$8$ |
$C_5^3:D_{20}$ |
5F1 |
1A |
$d^{8}f$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5G |
$5$ |
$16$ |
$C_5^2:C_{10}^2$ |
5G |
1A |
$d^{4}f$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5H |
$5$ |
$16$ |
$C_5^2:C_{10}^2$ |
5H |
1A |
$d^{4}g$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5I |
$5$ |
$16$ |
$C_5^2:C_{10}^2$ |
5I |
1A |
$d^{4}e^{8}f^{3}g$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5J |
$5$ |
$16$ |
$C_5^2:C_{10}^2$ |
5J |
1A |
$fg^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5K |
$5$ |
$16$ |
$C_5^3:C_{20}$ |
5K |
1A |
$d^{8}e^{6}f^{2}g^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5L |
$5$ |
$16$ |
$C_5^3:C_{20}$ |
5L |
1A |
$d^{8}e^{8}f^{2}g^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5M |
$5$ |
$16$ |
$C_5^3:C_{20}$ |
5M |
1A |
$d^{4}e^{6}f^{4}g$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5N |
$5$ |
$16$ |
$C_5^3:C_{20}$ |
5N |
1A |
$d^{6}e^{8}fg^{2}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5O |
$5$ |
$16$ |
$C_5^3:C_{20}$ |
5O |
1A |
$d^{4}e^{8}f^{3}g^{4}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5P1 |
$5$ |
$32$ |
$C_5^3\times C_{10}$ |
5P2 |
1A |
$d^{4}fg$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5P2 |
$5$ |
$32$ |
$C_5^3\times C_{10}$ |
5P1 |
1A |
$d^{8}f^{2}g^{2}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5Q1 |
$5$ |
$32$ |
$C_5^3\times C_{10}$ |
5Q2 |
1A |
$d^{4}fg^{3}$ |
| $(\SO(3,7)\times S_4^2).C_2^2$ |
5Q2 |
$5$ |
$32$ |
$C_5^3\times C_{10}$ |
5Q1 |
1A |
$d^{8}f^{2}g$ |
