This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_5 \times (C_{24}:C_2^4)$ | |
| Order: | \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) | |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) | |
| Automorphism group: | Group of order 98304 | |
| 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 | 5 | 6 | 8 | 10 | 12 | 15 | 20 | 24 | 30 | 40 | 60 | 120 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 127 | 2 | 64 | 4 | 62 | 64 | 508 | 32 | 8 | 256 | 32 | 248 | 256 | 128 | 128 | 1920 | |
| Conjugacy classes | 1 | 23 | 1 | 12 | 4 | 11 | 8 | 92 | 6 | 4 | 48 | 4 | 44 | 32 | 24 | 16 | 330 | |
| Divisions | data not computed | |||||||||||||||||
| Autjugacy classes | data not computed | |||||||||||||||||
| Dimension | 1 | 2 | 4 | 8 | |
|---|---|---|---|---|---|
| Irr. complex chars. | 160 | 120 | 40 | 10 | 330 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=g^{2}= \!\cdots\! \rangle}$
| |||||
| Aut. group: | $\Aut(C_{11}\times D_{24})$ | $\Aut(C_{33}:Q_{16})$ | ||||
Homology
| Abelianization: | $C_{2}^{4} \times C_{10} \simeq C_{2}^{5} \times C_{5}$ |
Subgroups
| Center: | $Z \simeq$ $C_2\times C_{10}$ | $G/Z \simeq$ $D_4\times D_6$ | |
| Commutator: | $G' \simeq$ $C_{12}$ | $G/G' \simeq$ $C_2^4\times C_{10}$ | |
| Frattini: | $\Phi \simeq$ $C_4$ | $G/\Phi \simeq$ $C_{15}:C_2^5$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{120}:C_2^3$ | $G/\operatorname{Fit} \simeq$ $C_2$ | |
| Radical: | $R \simeq$ $C_5 \times (C_{24}:C_2^4)$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2\times C_{30}$ | $G/S \simeq$ $C_2^2\times D_4$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_8:C_2^4$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ | ||
| Maximal subgroups: | $M_{2,1} \simeq$ $C_{120}:C_2^3$ | $G/M_{2,1} \simeq$ $C_2$ | |
| $M_{2,2} \simeq$ $C_{120}:C_2^3$ | $G/M_{2,2} \simeq$ $C_2$ | ||
| $M_{2,3} \simeq$ $C_{120}:C_2^3$ | $G/M_{2,3} \simeq$ $C_2$ | 2 normal subgroups | |
| $M_{2,4} \simeq$ $C_{120}:C_2^3$ | $G/M_{2,4} \simeq$ $C_2$ | 2 normal subgroups | |
| $M_{2,5} \simeq$ $C_{120}:C_2^3$ | $G/M_{2,5} \simeq$ $C_2$ | 2 normal subgroups | |
| $M_{2,6} \simeq$ $C_{120}:C_2^3$ | $G/M_{2,6} \simeq$ $C_2$ | 2 normal subgroups | |
| $M_{2,7} \simeq$ $C_{120}:C_2^3$ | $G/M_{2,7} \simeq$ $C_2$ | 16 normal subgroups | |
| $M_{2,8} \simeq$ $C_{60}.C_2^4$ | $G/M_{2,8} \simeq$ $C_2$ | ||
| $M_{2,9} \simeq$ $C_{60}.C_2^4$ | $G/M_{2,9} \simeq$ $C_2$ | ||
| $M_{2,10} \simeq$ $C_{120}:C_2^3$ | $G/M_{2,10} \simeq$ $C_2$ | ||
| $M_{2,11} \simeq$ $C_{60}:C_2^4$ | $G/M_{2,11} \simeq$ $C_2$ | ||
| $M_{2,12} \simeq$ $C_{30}.C_2^5$ | $G/M_{2,12} \simeq$ $C_2$ | ||
| $M_{3} \simeq$ $C_5 \times (C_8:C_2^4)$ | 3 subgroups in one conjugacy class | ||
| $M_{5} \simeq$ $C_{24}:C_2^4$ | $G/M_{5} \simeq$ $C_5$ | ||
| Maximal quotients: | $m_{2,1} \simeq$ $C_2$ | $G/m_{2,1} \simeq$ $C_{120}:C_2^3$ | 2 normal subgroups |
| $m_{2,2} \simeq$ $C_2$ | $G/m_{2,2} \simeq$ $C_{60}:C_2^4$ | ||
| $m_{3} \simeq$ $C_3$ | $G/m_{3} \simeq$ $C_5 \times (C_8:C_2^4)$ | ||
| $m_{5} \simeq$ $C_5$ | $G/m_{5} \simeq$ $C_{24}:C_2^4$ |