This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_3 \times (C_5 \rtimes (C_2\times C_4\times C_{16}))$ | |
| Order: | \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) | |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) | |
| Automorphism group: | Group of order 10240 | |
| Derived length: | $2$ |
This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group. Whether it is metacyclic, monomial, or rational has not been computed.
Group statistics
| Order | 1 | 2 | 3 | 4 | 5 | 6 | 8 | 10 | 12 | 15 | 16 | 20 | 24 | 30 | 40 | 48 | 60 | 80 | 120 | 240 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 23 | 2 | 104 | 4 | 46 | 128 | 12 | 208 | 8 | 256 | 16 | 256 | 24 | 32 | 512 | 32 | 64 | 64 | 128 | 1920 | |
| Conjugacy classes | 1 | 7 | 2 | 24 | 1 | 14 | 32 | 3 | 48 | 2 | 64 | 4 | 64 | 6 | 8 | 128 | 8 | 16 | 16 | 32 | 480 | |
| Divisions | data not computed | |||||||||||||||||||||
| Autjugacy classes | data not computed | |||||||||||||||||||||
| Dimension | 1 | 4 | |
|---|---|---|---|
| Irr. complex chars. | 384 | 96 | 480 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i \mid c^{2}=e^{2}=g^{2}=h^{3}=i^{5}=[a,b]= \!\cdots\! \rangle}$
| |||||
| Aut. group: | $\Aut(D_5\times C_{119})$ | $\Aut(D_5\times C_{153})$ | ||||
Homology
| Abelianization: | $C_{2} \times C_{4} \times C_{48} \simeq C_{2} \times C_{4} \times C_{16} \times C_{3}$ |
Subgroups
| Center: | $Z \simeq$ $C_2\times C_{48}$ | $G/Z \simeq$ $F_5$ | |
| Commutator: | $G' \simeq$ $C_5$ | $G/G' \simeq$ $C_2\times C_4\times C_{48}$ | |
| Frattini: | $\Phi \simeq$ $C_8$ | $G/\Phi \simeq$ $D_{10}:C_{12}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_2\times C_{240}$ | $G/\operatorname{Fit} \simeq$ $C_4$ | |
| Radical: | $R \simeq$ $C_3 \times (C_5 \rtimes (C_2\times C_4\times C_{16}))$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2\times C_{30}$ | $G/S \simeq$ $C_4\times C_8$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2\times C_4\times C_{16}$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ | ||
| Maximal subgroups: | $M_{2,1} \simeq$ $D_{10}\times C_{48}$ | $G/M_{2,1} \simeq$ $C_2$ | |
| $M_{2,2} \simeq$ $D_{10}:C_{48}$ | $G/M_{2,2} \simeq$ $C_2$ | ||
| $M_{2,3} \simeq$ $C_{60}.C_4^2$ | $G/M_{2,3} \simeq$ $C_2$ | ||
| $M_{2,4} \simeq$ $F_5\times C_{48}$ | $G/M_{2,4} \simeq$ $C_2$ | 4 normal subgroups | |
| $M_{3} \simeq$ $C_5 \rtimes (C_2\times C_4\times C_{16})$ | $G/M_{3} \simeq$ $C_3$ | ||
| $M_{5} \simeq$ $C_2\times C_4\times C_{48}$ | 5 subgroups in one conjugacy class | ||
| Maximal quotients: | $m_{2,1} \simeq$ $C_2$ | $G/m_{2,1} \simeq$ $C_{60}.C_4^2$ | |
| $m_{2,2} \simeq$ $C_2$ | $G/m_{2,2} \simeq$ $F_5\times C_{48}$ | 2 normal subgroups | |
| $m_{3} \simeq$ $C_3$ | $G/m_{3} \simeq$ $C_5 \rtimes (C_2\times C_4\times C_{16})$ | ||
| $m_{5} \simeq$ $C_5$ | $G/m_{5} \simeq$ $C_2\times C_4\times C_{48}$ |