This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_3 \rtimes (\OD_{16}:C_2^4)$ | |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) | |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) | |
| Automorphism group: | Group of order 7077888 | |
| 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 | 24 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 127 | 2 | 128 | 62 | 256 | 64 | 128 | 768 | |
| Conjugacy classes | 1 | 39 | 1 | 40 | 19 | 80 | 20 | 40 | 240 | |
| Divisions | data not computed | |||||||||
| Autjugacy classes | data not computed | |||||||||
| Dimension | 1 | 2 | 4 | |
|---|---|---|---|---|
| Irr. complex chars. | 128 | 96 | 16 | 240 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i \mid b^{2}=c^{2}=d^{2}=e^{2}=f^{2}=h^{2}= \!\cdots\! \rangle}$
| |||||
| Aut. group: | $\Aut(C_{48}.D_4)$ | $\Aut(S_3\times \OD_{64})$ | $\Aut(C_6:C_{96})$ | $\Aut(C_{12}.C_{48})$ | ||
Homology
| Abelianization: | $C_{2}^{5} \times C_{4} $ |
Subgroups
| Center: | $Z \simeq$ $C_2^2\times C_8$ | $G/Z \simeq$ $C_2\times D_6$ | |
| Commutator: | $G' \simeq$ $C_6$ | $G/G' \simeq$ $C_2^5\times C_4$ | |
| Frattini: | $\Phi \simeq$ $C_4$ | $G/\Phi \simeq$ $D_6\times C_2^4$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{12}.C_2^5$ | $G/\operatorname{Fit} \simeq$ $C_2$ | |
| Radical: | $R \simeq$ $C_3 \rtimes (\OD_{16}:C_2^4)$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^2\times C_6$ | $G/S \simeq$ $C_2^3\times C_4$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $\OD_{16}:C_2^4$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3$ | ||
| Maximal subgroups: | $M_{2,1} \simeq$ $C_{24}:C_2^4$ | $G/M_{2,1} \simeq$ $C_2$ | 3 normal subgroups |
| $M_{2,2} \simeq$ $C_{12}.C_2^5$ | $G/M_{2,2} \simeq$ $C_2$ | 3 normal subgroups | |
| $M_{2,3} \simeq$ $C_{24}:C_2^4$ | $G/M_{2,3} \simeq$ $C_2$ | 3 normal subgroups | |
| $M_{2,4} \simeq$ $C_{12}.C_2^5$ | $G/M_{2,4} \simeq$ $C_2$ | 3 normal subgroups | |
| $M_{2,5} \simeq$ $C_{12}.C_2^5$ | $G/M_{2,5} \simeq$ $C_2$ | 48 normal subgroups | |
| $M_{2,6} \simeq$ $C_{12}.C_2^5$ | $G/M_{2,6} \simeq$ $C_2$ | ||
| $M_{2,7} \simeq$ $C_{12}.C_2^5$ | $G/M_{2,7} \simeq$ $C_2$ | ||
| $M_{2,8} \simeq$ $D_{12}:C_2^4$ | $G/M_{2,8} \simeq$ $C_2$ | ||
| $M_{3} \simeq$ $\OD_{16}:C_2^4$ | 3 subgroups in one conjugacy class | ||
| Maximal quotients: | $m_{2,1} \simeq$ $C_2$ | $G/m_{2,1} \simeq$ $C_{12}.C_2^5$ | 6 normal subgroups |
| $m_{2,2} \simeq$ $C_2$ | $G/m_{2,2} \simeq$ $C_{12}:C_2^5$ | ||
| $m_{3} \simeq$ $C_3$ | $G/m_{3} \simeq$ $\OD_{16}:C_2^4$ |