This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{2} \times C_{4} \times C_{8640}$ |
| Order: | \(138240\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| Automorphism group: | Group of order \(113246208\)\(\medspace = 2^{22} \cdot 3^{3} \) |
| Outer automorphisms: | Group of order \(113246208\)\(\medspace = 2^{22} \cdot 3^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary). Whether it is metacyclic or rational has not been computed.
Group statistics
| Order | 1 | 2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 12 | 15 | 16 | 18 | 20 | 24 | 27 | 30 | 32 | 36 | 40 | 45 | 48 | 54 | 60 | 64 | 72 | 80 | 90 | 96 | 108 | 120 | 135 | 144 | 160 | 180 | 192 | 216 | 240 | 270 | 288 | 320 | 360 | 432 | 480 | 540 | 576 | 720 | 864 | 960 | 1080 | 1440 | 1728 | 2160 | 2880 | 4320 | 8640 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 15 | 2 | 48 | 4 | 30 | 64 | 6 | 60 | 96 | 8 | 128 | 90 | 192 | 128 | 18 | 120 | 256 | 288 | 256 | 24 | 256 | 270 | 384 | 512 | 384 | 512 | 360 | 512 | 864 | 512 | 72 | 768 | 1024 | 1152 | 1024 | 1152 | 1024 | 1080 | 1536 | 2048 | 1536 | 2304 | 2048 | 3456 | 3072 | 3072 | 4608 | 4096 | 4608 | 6144 | 9216 | 9216 | 12288 | 18432 | 36864 | 138240 |
| Conjugacy classes | 1 | 15 | 2 | 48 | 4 | 30 | 64 | 6 | 60 | 96 | 8 | 128 | 90 | 192 | 128 | 18 | 120 | 256 | 288 | 256 | 24 | 256 | 270 | 384 | 512 | 384 | 512 | 360 | 512 | 864 | 512 | 72 | 768 | 1024 | 1152 | 1024 | 1152 | 1024 | 1080 | 1536 | 2048 | 1536 | 2304 | 2048 | 3456 | 3072 | 3072 | 4608 | 4096 | 4608 | 6144 | 9216 | 9216 | 12288 | 18432 | 36864 | 138240 |
| Divisions | data not computed | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Autjugacy classes | data not computed | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 138240 | 138240 |
Constructions
| Rank: | $4$ |
| Inequivalent generating quadruples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{2} \times C_{4} \times C_{64} \times C_{27} \times C_{5}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{2} \times C_{4} \times C_{8640}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{2} \times C_{4} \times C_{8640}$ | |
| Frattini: | $\Phi \simeq$ $C_2\times C_{288}$ | $G/\Phi \simeq$ $C_2^3\times C_{30}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{2} \times C_{4} \times C_{8640}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{2} \times C_{4} \times C_{8640}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^3\times C_{30}$ | $G/S \simeq$ $C_2\times C_{288}$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_{2}^{2} \times C_{4} \times C_{64}$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_{27}$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ |