This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_3^5 . C_3^2$ | |
| Order: | \(2187\)\(\medspace = 3^{7} \) | |
| Exponent: | \(9\)\(\medspace = 3^{2} \) | |
| Automorphism group: | Group of order 696241109191680 | |
| Nilpotency class: | $2$ | |
| Derived length: | $2$ |
This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is metacyclic, monomial, or rational has not been computed.
Group statistics
| Order | 1 | 3 | 9 | ||
|---|---|---|---|---|---|
| Elements | 1 | 728 | 1458 | 2187 | |
| Conjugacy classes | 1 | 404 | 486 | 891 | |
| Divisions | data not computed | ||||
| Autjugacy classes | data not computed | ||||
| Dimension | 1 | 3 | |
|---|---|---|---|
| Irr. complex chars. | 729 | 162 | 891 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g \mid b^{3}=c^{3}=d^{3}=e^{3}=f^{3}=g^{3}=[a,c]= \!\cdots\! \rangle}$
| |||||
Homology
| Abelianization: | $C_{3}^{6} $ |
Subgroups
| Center: | $Z \simeq$ $C_3^5$ | $G/Z \simeq$ $C_3^2$ | |
| Commutator: | $G' \simeq$ $C_3$ | $G/G' \simeq$ $C_3^6$ | |
| Frattini: | $\Phi \simeq$ $C_3$ | $G/\Phi \simeq$ $C_3^6$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_3^5 . C_3^2$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_3^5 . C_3^2$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_3^5$ | $G/S \simeq$ $C_3^2$ | |
| Maximal subgroups: | $M_{3,1} \simeq$ $C_3^4\times C_9$ | $G/M_{3,1} \simeq$ $C_3$ | 3 normal subgroups |
| $M_{3,2} \simeq$ $C_9:C_3^4$ | $G/M_{3,2} \simeq$ $C_3$ | 360 normal subgroups | |
| $M_{3,3} \simeq$ $C_3^6$ | $G/M_{3,3} \simeq$ $C_3$ | ||
| Maximal quotients: | $m_{3,1} \simeq$ $C_3$ | $G/m_{3,1} \simeq$ $C_9:C_3^4$ | 120 normal subgroups |
| $m_{3,2} \simeq$ $C_3$ | $G/m_{3,2} \simeq$ $C_3^6$ |