This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_3^2 \rtimes (C_2^4:D_4)$ | |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) | |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) | |
| Automorphism group: | Group of order 1179648 | |
| Derived length: | $2$ |
This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian. Whether it is metacyclic, monomial, or rational has not been computed.
Group statistics
| Order | 1 | 2 | 3 | 4 | 6 | 12 | ||
|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 159 | 8 | 96 | 552 | 336 | 1152 | |
| Conjugacy classes | 1 | 43 | 5 | 12 | 149 | 42 | 252 | |
| Divisions | data not computed | |||||||
| Autjugacy classes | data not computed | |||||||
| Dimension | 1 | 2 | 4 | |
|---|---|---|---|---|
| Irr. complex chars. | 96 | 120 | 36 | 252 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
| |||||
| Aut. group: | $\Aut(C_{12}:C_{28})$ | $\Aut(C_{12}:C_{28})$ | $\Aut(C_{42}.D_4)$ | $\Aut(D_6\times C_{28})$ | all 12 | |
Homology
| Abelianization: | $C_{2}^{4} \times C_{6} \simeq C_{2}^{5} \times C_{3}$ |
Subgroups
| Center: | $Z \simeq$ $C_2^2\times C_6$ | $G/Z \simeq$ $C_2^2\times D_6$ | |
| Commutator: | $G' \simeq$ $C_2\times C_6$ | $G/G' \simeq$ $C_2^4\times C_6$ | |
| Frattini: | $\Phi \simeq$ $C_2^2$ | $G/\Phi \simeq$ $C_6^2:C_2^3$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_2^4:C_6^2$ | $G/\operatorname{Fit} \simeq$ $C_2$ | |
| Radical: | $R \simeq$ $C_3^2 \rtimes (C_2^4:D_4)$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2\times C_6^2$ | $G/S \simeq$ $C_2^4$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^4:D_4$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3^2$ | ||
| Maximal subgroups: | $M_{2,1} \simeq$ $C_6^2.C_2^4$ | $G/M_{2,1} \simeq$ $C_2$ | 3 normal subgroups |
| $M_{2,2} \simeq$ $C_6^2.C_2^4$ | $G/M_{2,2} \simeq$ $C_2$ | 3 normal subgroups | |
| $M_{2,3} \simeq$ $C_6^2.C_2^4$ | $G/M_{2,3} \simeq$ $C_2$ | 16 normal subgroups | |
| $M_{2,4} \simeq$ $C_6^2.C_2^4$ | $G/M_{2,4} \simeq$ $C_2$ | 3 normal subgroups | |
| $M_{2,5} \simeq$ $C_6^2.C_2^4$ | $G/M_{2,5} \simeq$ $C_2$ | ||
| $M_{2,6} \simeq$ $C_2^4:C_6^2$ | $G/M_{2,6} \simeq$ $C_2$ | ||
| $M_{2,7} \simeq$ $C_6^2:C_2^4$ | $G/M_{2,7} \simeq$ $C_2$ | 3 normal subgroups | |
| $M_{2,8} \simeq$ $C_6^2:C_2^4$ | $G/M_{2,8} \simeq$ $C_2$ | ||
| $M_{3,1} \simeq$ $C_2^5:D_6$ | $G/M_{3,1} \simeq$ $C_3$ | ||
| $M_{3,2} \simeq$ $C_2^6:C_6$ | 3 subgroups in one conjugacy class | ||
| Maximal quotients: | $m_{2,1} \simeq$ $C_2$ | $G/m_{2,1} \simeq$ $C_6^2.C_2^4$ | 4 normal subgroups |
| $m_{2,2} \simeq$ $C_2$ | $G/m_{2,2} \simeq$ $C_6^2:C_2^4$ | 3 normal subgroups | |
| $m_{3,1} \simeq$ $C_3$ | $G/m_{3,1} \simeq$ $C_2^5:D_6$ | ||
| $m_{3,2} \simeq$ $C_3$ | $G/m_{3,2} \simeq$ $C_2^6:C_6$ |