This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_2^6 \times S_4$ | |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) | |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) | |
| Automorphism group: | Group of order 30963778191360 | |
| Derived length: | $3$ |
This group is nonabelian and solvable. Whether it is metacyclic, monomial, or rational has not been computed.
Group statistics
| Order | 1 | 2 | 3 | 4 | 6 | ||
|---|---|---|---|---|---|---|---|
| Elements | 1 | 639 | 8 | 384 | 504 | 1536 | |
| Conjugacy classes | 1 | 191 | 1 | 64 | 63 | 320 | |
| Divisions | data not computed | ||||||
| Autjugacy classes | data not computed | ||||||
| Dimension | 1 | 2 | 3 | |
|---|---|---|---|---|
| Irr. complex chars. | 128 | 64 | 128 | 320 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i, j \mid a^{2}=b^{3}=c^{2}=d^{2}=e^{2}= \!\cdots\! \rangle}$
| |||||
| Aut. group: | $\Aut(C_2^2.\GL(2,\mathbb{Z}/4))$ | $\Aut(C_2^2.\GL(2,\mathbb{Z}/4))$ | $\Aut(C_2^2.\GL(2,\mathbb{Z}/4))$ | $\Aut(C_2^2.\GL(2,\mathbb{Z}/4))$ | all 67 | |
Homology
| Abelianization: | $C_{2}^{7} $ |
Subgroups
| Center: | $Z \simeq$ $C_2^6$ | $G/Z \simeq$ $S_4$ | |
| Commutator: | $G' \simeq$ $A_4$ | $G/G' \simeq$ $C_2^7$ | |
| Frattini: | $\Phi \simeq$ $C_1$ | $G/\Phi \simeq$ $C_2^6 \times S_4$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_2^8$ | $G/\operatorname{Fit} \simeq$ $S_3$ | |
| Radical: | $R \simeq$ $C_2^6 \times S_4$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^8$ | $G/S \simeq$ $S_3$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^7 . C_2^2$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3$ | ||
| Maximal subgroups: | $M_{2,1} \simeq$ $C_2^8 \rtimes C_3$ | $G/M_{2,1} \simeq$ $C_2$ | |
| $M_{2,2} \simeq$ $S_4\times C_2^5$ | $G/M_{2,2} \simeq$ $C_2$ | 126 normal subgroups | |
| $M_{3} \simeq$ $C_2^7 . C_2^2$ | 3 subgroups in one conjugacy class | ||
| $M_{4} \simeq$ $D_6\times C_2^5$ | 4 subgroups in one conjugacy class | ||
| Maximal quotients: | $m_{2} \simeq$ $C_2$ | $G/m_{2} \simeq$ $S_4\times C_2^5$ | 63 normal subgroups |
| $m_{4} \simeq$ $C_2^2$ | $G/m_{4} \simeq$ $D_6\times C_2^5$ |