This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{3} \times C_{27360}$ | |
| Order: | \(218880\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 19 \) | |
| Exponent: | \(27360\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 19 \) | |
| Automorphism group: | Group of order 74317824 | |
| 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 | 19 | 20 | 24 | 30 | 32 | 36 | 38 | 40 | 45 | 48 | 57 | 60 | 72 | 76 | 80 | 90 | 95 | 96 | 114 | 120 | 144 | 152 | 160 | 171 | 180 | 190 | 228 | 240 | 285 | 288 | 304 | 342 | 360 | 380 | 456 | 480 | 570 | 608 | 684 | 720 | 760 | 855 | 912 | 1140 | 1368 | 1440 | 1520 | 1710 | 1824 | 2280 | 2736 | 3040 | 3420 | 4560 | 5472 | 6840 | 9120 | 13680 | 27360 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 15 | 2 | 16 | 4 | 30 | 32 | 6 | 60 | 32 | 8 | 64 | 90 | 18 | 64 | 64 | 120 | 128 | 96 | 270 | 128 | 24 | 128 | 36 | 128 | 192 | 288 | 256 | 360 | 72 | 256 | 540 | 256 | 384 | 576 | 512 | 108 | 384 | 1080 | 576 | 512 | 144 | 768 | 1152 | 1620 | 768 | 1152 | 1152 | 1024 | 2160 | 2304 | 1728 | 1536 | 2304 | 432 | 2304 | 2304 | 3456 | 3072 | 4608 | 6480 | 4608 | 4608 | 6912 | 9216 | 6912 | 9216 | 13824 | 13824 | 18432 | 27648 | 55296 | 218880 | |
| Conjugacy classes | 1 | 15 | 2 | 16 | 4 | 30 | 32 | 6 | 60 | 32 | 8 | 64 | 90 | 18 | 64 | 64 | 120 | 128 | 96 | 270 | 128 | 24 | 128 | 36 | 128 | 192 | 288 | 256 | 360 | 72 | 256 | 540 | 256 | 384 | 576 | 512 | 108 | 384 | 1080 | 576 | 512 | 144 | 768 | 1152 | 1620 | 768 | 1152 | 1152 | 1024 | 2160 | 2304 | 1728 | 1536 | 2304 | 432 | 2304 | 2304 | 3456 | 3072 | 4608 | 6480 | 4608 | 4608 | 6912 | 9216 | 6912 | 9216 | 13824 | 13824 | 18432 | 27648 | 55296 | 218880 | |
| Divisions | data not computed | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Autjugacy classes | data not computed | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 218880 | 218880 |
Constructions
| Rank: | $4$ |
| Inequivalent generating quadruples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{3} \times C_{32} \times C_{9} \times C_{5} \times C_{19}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{3} \times C_{27360}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{3} \times C_{27360}$ | |
| Frattini: | $\Phi \simeq$ $C_{48}$ | $G/\Phi \simeq$ $C_{2}^{2} \times C_{1140}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{3} \times C_{27360}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{3} \times C_{27360}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_{2}^{2} \times C_{1140}$ | $G/S \simeq$ $C_{48}$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^3\times C_{32}$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_9$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ | ||
| 19-Sylow subgroup: | $P_{19} \simeq$ $C_{19}$ |