This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_3 \rtimes (C_4\times D_{32})$ | |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) | |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) | |
| Automorphism group: | Group of order 49152 | |
| Derived length: | $2$ |
This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian. Whether it is metacyclic, monomial, or rational has not been computed.
Group statistics
| Order | 1 | 2 | 3 | 4 | 6 | 8 | 12 | 16 | 24 | 32 | 48 | 96 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 195 | 2 | 204 | 6 | 16 | 24 | 32 | 32 | 64 | 64 | 128 | 768 | |
| Conjugacy classes | 1 | 7 | 1 | 12 | 3 | 8 | 12 | 16 | 16 | 32 | 32 | 64 | 204 | |
| Divisions | data not computed | |||||||||||||
| Autjugacy classes | data not computed | |||||||||||||
| Dimension | 1 | 2 | |
|---|---|---|---|
| Irr. complex chars. | 16 | 188 | 204 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i \mid b^{2}=e^{2}=h^{2}=i^{3}=[a,c]=[a,e]= \!\cdots\! \rangle}$
| |||||
Homology
| Abelianization: | $C_{2}^{2} \times C_{4} $ |
Subgroups
| Center: | $Z \simeq$ $C_2\times C_4$ | $G/Z \simeq$ $D_{48}$ | |
| Commutator: | $G' \simeq$ $C_{48}$ | $G/G' \simeq$ $C_2^2\times C_4$ | |
| Frattini: | $\Phi \simeq$ $C_2\times C_{16}$ | $G/\Phi \simeq$ $C_2\times D_6$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_4\times C_{96}$ | $G/\operatorname{Fit} \simeq$ $C_2$ | |
| Radical: | $R \simeq$ $C_3 \rtimes (C_4\times D_{32})$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2\times C_6$ | $G/S \simeq$ $C_2\times D_{16}$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_4\times D_{32}$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3$ | ||
| Maximal subgroups: | $M_{2,1} \simeq$ $C_4\times D_{48}$ | $G/M_{2,1} \simeq$ $C_2$ | 2 normal subgroups |
| $M_{2,2} \simeq$ $C_{96}:C_4$ | $G/M_{2,2} \simeq$ $C_2$ | ||
| $M_{2,3} \simeq$ $D_{48}:C_4$ | $G/M_{2,3} \simeq$ $C_2$ | 2 normal subgroups | |
| $M_{2,4} \simeq$ $C_2\times D_{96}$ | $G/M_{2,4} \simeq$ $C_2$ | ||
| $M_{2,5} \simeq$ $C_4\times C_{96}$ | $G/M_{2,5} \simeq$ $C_2$ | ||
| $M_{3} \simeq$ $C_4\times D_{32}$ | 3 subgroups in one conjugacy class | ||
| Maximal quotients: | $m_{2,1} \simeq$ $C_2$ | $G/m_{2,1} \simeq$ $C_4\times D_{48}$ | |
| $m_{2,2} \simeq$ $C_2$ | $G/m_{2,2} \simeq$ $C_2\times D_{96}$ | ||
| $m_{2,3} \simeq$ $C_2$ | $G/m_{2,3} \simeq$ $D_{96}:C_2$ | ||
| $m_{3} \simeq$ $C_3$ | $G/m_{3} \simeq$ $C_4\times D_{32}$ |