This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $(C_2^3.C_2^5) \rtimes C_3$ | |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) | |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) | |
| Automorphism group: | Group of order 73728 | |
| Derived length: | $2$ |
This group is nonabelian and metabelian (hence solvable). Whether it is metacyclic, monomial, or rational has not been computed.
Group statistics
| Order | 1 | 2 | 3 | 4 | 6 | 12 | ||
|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 31 | 32 | 224 | 224 | 256 | 768 | |
| Conjugacy classes | 1 | 15 | 2 | 32 | 14 | 16 | 80 | |
| Divisions | data not computed | |||||||
| Autjugacy classes | data not computed | |||||||
| Dimension | 1 | 3 | 6 | |
|---|---|---|---|---|
| Irr. complex chars. | 48 | 16 | 16 | 80 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i \mid a^{3}=e^{2}=f^{2}=g^{2}=h^{2}=i^{2}= \!\cdots\! \rangle}$
| |||||
Homology
| Abelianization: | $C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$ |
Subgroups
| Center: | $Z \simeq$ $C_2^3$ | $G/Z \simeq$ $C_4^2:C_6$ | |
| Commutator: | $G' \simeq$ $C_4^2$ | $G/G' \simeq$ $C_2^2\times C_{12}$ | |
| Frattini: | $\Phi \simeq$ $C_2^3$ | $G/\Phi \simeq$ $C_2^3\times A_4$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_2^3.C_2^5$ | $G/\operatorname{Fit} \simeq$ $C_3$ | |
| Radical: | $R \simeq$ $(C_2^3.C_2^5) \rtimes C_3$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^5$ | $G/S \simeq$ $C_2\times A_4$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^3.C_2^5$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3$ | ||
| Maximal subgroups: | $M_{2,1} \simeq$ $C_2^5.A_4$ | $G/M_{2,1} \simeq$ $C_2$ | |
| $M_{2,2} \simeq$ $C_2\times C_4^2:C_{12}$ | $G/M_{2,2} \simeq$ $C_2$ | 6 normal subgroups | |
| $M_{3} \simeq$ $C_2^3.C_2^5$ | $G/M_{3} \simeq$ $C_3$ | ||
| $M_{4} \simeq$ $C_2^4:C_{12}$ | 4 subgroups in one conjugacy class | ||
| Maximal quotients: | $m_{2,1} \simeq$ $C_2$ | $G/m_{2,1} \simeq$ $C_2^5.A_4$ | |
| $m_{2,2} \simeq$ $C_2$ | $G/m_{2,2} \simeq$ $C_2\times C_4^2:C_{12}$ | 6 normal subgroups | |
| $m_{4} \simeq$ $C_2^2$ | $G/m_{4} \simeq$ $C_2^4:C_{12}$ |