metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C41⋊2C8, C82.C4, Dic41.2C2, C2.(C41⋊C4), SmallGroup(328,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C41⋊2C8 |
Generators and relations for C41⋊2C8
G = < a,b | a41=b8=1, bab-1=a9 >
Character table of C41⋊2C8
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 41A | 41B | 41C | 41D | 41E | 41F | 41G | 41H | 41I | 41J | 82A | 82B | 82C | 82D | 82E | 82F | 82G | 82H | 82I | 82J | |
| size | 1 | 1 | 41 | 41 | 41 | 41 | 41 | 41 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | i | -i | ζ87 | ζ85 | ζ83 | ζ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ6 | 1 | -1 | -i | i | ζ8 | ζ83 | ζ85 | ζ87 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ7 | 1 | -1 | -i | i | ζ85 | ζ87 | ζ8 | ζ83 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ8 | 1 | -1 | i | -i | ζ83 | ζ8 | ζ87 | ζ85 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 8 |
| ρ9 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4140+ζ4132+ζ419+ζ41 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4140+ζ4132+ζ419+ζ41 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4138+ζ4127+ζ4114+ζ413 | orthogonal lifted from C41⋊C4 |
| ρ10 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4140+ζ4132+ζ419+ζ41 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4140+ζ4132+ζ419+ζ41 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4139+ζ4123+ζ4118+ζ412 | orthogonal lifted from C41⋊C4 |
| ρ11 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4137+ζ4136+ζ415+ζ414 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4130+ζ4124+ζ4117+ζ4111 | orthogonal lifted from C41⋊C4 |
| ρ12 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4140+ζ4132+ζ419+ζ41 | orthogonal lifted from C41⋊C4 |
| ρ13 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4140+ζ4132+ζ419+ζ41 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4140+ζ4132+ζ419+ζ41 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4137+ζ4136+ζ415+ζ414 | orthogonal lifted from C41⋊C4 |
| ρ14 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4140+ζ4132+ζ419+ζ41 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4140+ζ4132+ζ419+ζ41 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4134+ζ4122+ζ4119+ζ417 | orthogonal lifted from C41⋊C4 |
| ρ15 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4137+ζ4136+ζ415+ζ414 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4140+ζ4132+ζ419+ζ41 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4137+ζ4136+ζ415+ζ414 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4135+ζ4128+ζ4113+ζ416 | orthogonal lifted from C41⋊C4 |
| ρ16 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4137+ζ4136+ζ415+ζ414 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4137+ζ4136+ζ415+ζ414 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4129+ζ4126+ζ4115+ζ4112 | orthogonal lifted from C41⋊C4 |
| ρ17 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4137+ζ4136+ζ415+ζ414 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4137+ζ4136+ζ415+ζ414 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4125+ζ4121+ζ4120+ζ4116 | orthogonal lifted from C41⋊C4 |
| ρ18 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4137+ζ4136+ζ415+ζ414 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4137+ζ4136+ζ415+ζ414 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4133+ζ4131+ζ4110+ζ418 | orthogonal lifted from C41⋊C4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4137+ζ4136+ζ415+ζ414 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4125-ζ4121-ζ4120-ζ4116 | symplectic faithful, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4140+ζ4132+ζ419+ζ41 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4134-ζ4122-ζ4119-ζ417 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4137+ζ4136+ζ415+ζ414 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4129-ζ4126-ζ4115-ζ4112 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4140+ζ4132+ζ419+ζ41 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4137-ζ4136-ζ415-ζ414 | symplectic faithful, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4137+ζ4136+ζ415+ζ414 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4133-ζ4131-ζ4110-ζ418 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4130-ζ4124-ζ4117-ζ4111 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4140+ζ4132+ζ419+ζ41 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4137+ζ4136+ζ415+ζ414 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4125+ζ4121+ζ4120+ζ4116 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4138-ζ4127-ζ4114-ζ413 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4140+ζ4132+ζ419+ζ41 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4138+ζ4127+ζ4114+ζ413 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4139-ζ4123-ζ4118-ζ412 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4134+ζ4122+ζ4119+ζ417 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4137+ζ4136+ζ415+ζ414 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4140+ζ4132+ζ419+ζ41 | -ζ4140-ζ4132-ζ419-ζ41 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4135-ζ4128-ζ4113-ζ416 | symplectic faithful, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ4125+ζ4121+ζ4120+ζ4116 | ζ4133+ζ4131+ζ4110+ζ418 | ζ4138+ζ4127+ζ4114+ζ413 | ζ4135+ζ4128+ζ4113+ζ416 | ζ4129+ζ4126+ζ4115+ζ4112 | ζ4139+ζ4123+ζ4118+ζ412 | ζ4130+ζ4124+ζ4117+ζ4111 | ζ4137+ζ4136+ζ415+ζ414 | ζ4140+ζ4132+ζ419+ζ41 | ζ4134+ζ4122+ζ4119+ζ417 | -ζ4134-ζ4122-ζ4119-ζ417 | -ζ4125-ζ4121-ζ4120-ζ4116 | -ζ4133-ζ4131-ζ4110-ζ418 | -ζ4138-ζ4127-ζ4114-ζ413 | -ζ4135-ζ4128-ζ4113-ζ416 | -ζ4129-ζ4126-ζ4115-ζ4112 | -ζ4139-ζ4123-ζ4118-ζ412 | -ζ4130-ζ4124-ζ4117-ζ4111 | -ζ4137-ζ4136-ζ415-ζ414 | -ζ4140-ζ4132-ζ419-ζ41 | symplectic faithful, Schur index 2 |
►Regular action on 328 points
Generators in S328
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41)(42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82)(83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123)(124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164)(165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205)(206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246)(247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287)(288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328)
(1 290 124 245 58 282 83 198)(2 322 164 213 59 273 123 166)(3 313 163 222 60 264 122 175)(4 304 162 231 61 255 121 184)(5 295 161 240 62 287 120 193)(6 327 160 208 63 278 119 202)(7 318 159 217 64 269 118 170)(8 309 158 226 65 260 117 179)(9 300 157 235 66 251 116 188)(10 291 156 244 67 283 115 197)(11 323 155 212 68 274 114 165)(12 314 154 221 69 265 113 174)(13 305 153 230 70 256 112 183)(14 296 152 239 71 247 111 192)(15 328 151 207 72 279 110 201)(16 319 150 216 73 270 109 169)(17 310 149 225 74 261 108 178)(18 301 148 234 75 252 107 187)(19 292 147 243 76 284 106 196)(20 324 146 211 77 275 105 205)(21 315 145 220 78 266 104 173)(22 306 144 229 79 257 103 182)(23 297 143 238 80 248 102 191)(24 288 142 206 81 280 101 200)(25 320 141 215 82 271 100 168)(26 311 140 224 42 262 99 177)(27 302 139 233 43 253 98 186)(28 293 138 242 44 285 97 195)(29 325 137 210 45 276 96 204)(30 316 136 219 46 267 95 172)(31 307 135 228 47 258 94 181)(32 298 134 237 48 249 93 190)(33 289 133 246 49 281 92 199)(34 321 132 214 50 272 91 167)(35 312 131 223 51 263 90 176)(36 303 130 232 52 254 89 185)(37 294 129 241 53 286 88 194)(38 326 128 209 54 277 87 203)(39 317 127 218 55 268 86 171)(40 308 126 227 56 259 85 180)(41 299 125 236 57 250 84 189)
G:=sub<Sym(328)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82)(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)(165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205)(206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246)(247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287)(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328), (1,290,124,245,58,282,83,198)(2,322,164,213,59,273,123,166)(3,313,163,222,60,264,122,175)(4,304,162,231,61,255,121,184)(5,295,161,240,62,287,120,193)(6,327,160,208,63,278,119,202)(7,318,159,217,64,269,118,170)(8,309,158,226,65,260,117,179)(9,300,157,235,66,251,116,188)(10,291,156,244,67,283,115,197)(11,323,155,212,68,274,114,165)(12,314,154,221,69,265,113,174)(13,305,153,230,70,256,112,183)(14,296,152,239,71,247,111,192)(15,328,151,207,72,279,110,201)(16,319,150,216,73,270,109,169)(17,310,149,225,74,261,108,178)(18,301,148,234,75,252,107,187)(19,292,147,243,76,284,106,196)(20,324,146,211,77,275,105,205)(21,315,145,220,78,266,104,173)(22,306,144,229,79,257,103,182)(23,297,143,238,80,248,102,191)(24,288,142,206,81,280,101,200)(25,320,141,215,82,271,100,168)(26,311,140,224,42,262,99,177)(27,302,139,233,43,253,98,186)(28,293,138,242,44,285,97,195)(29,325,137,210,45,276,96,204)(30,316,136,219,46,267,95,172)(31,307,135,228,47,258,94,181)(32,298,134,237,48,249,93,190)(33,289,133,246,49,281,92,199)(34,321,132,214,50,272,91,167)(35,312,131,223,51,263,90,176)(36,303,130,232,52,254,89,185)(37,294,129,241,53,286,88,194)(38,326,128,209,54,277,87,203)(39,317,127,218,55,268,86,171)(40,308,126,227,56,259,85,180)(41,299,125,236,57,250,84,189)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82)(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)(165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205)(206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246)(247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287)(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328), (1,290,124,245,58,282,83,198)(2,322,164,213,59,273,123,166)(3,313,163,222,60,264,122,175)(4,304,162,231,61,255,121,184)(5,295,161,240,62,287,120,193)(6,327,160,208,63,278,119,202)(7,318,159,217,64,269,118,170)(8,309,158,226,65,260,117,179)(9,300,157,235,66,251,116,188)(10,291,156,244,67,283,115,197)(11,323,155,212,68,274,114,165)(12,314,154,221,69,265,113,174)(13,305,153,230,70,256,112,183)(14,296,152,239,71,247,111,192)(15,328,151,207,72,279,110,201)(16,319,150,216,73,270,109,169)(17,310,149,225,74,261,108,178)(18,301,148,234,75,252,107,187)(19,292,147,243,76,284,106,196)(20,324,146,211,77,275,105,205)(21,315,145,220,78,266,104,173)(22,306,144,229,79,257,103,182)(23,297,143,238,80,248,102,191)(24,288,142,206,81,280,101,200)(25,320,141,215,82,271,100,168)(26,311,140,224,42,262,99,177)(27,302,139,233,43,253,98,186)(28,293,138,242,44,285,97,195)(29,325,137,210,45,276,96,204)(30,316,136,219,46,267,95,172)(31,307,135,228,47,258,94,181)(32,298,134,237,48,249,93,190)(33,289,133,246,49,281,92,199)(34,321,132,214,50,272,91,167)(35,312,131,223,51,263,90,176)(36,303,130,232,52,254,89,185)(37,294,129,241,53,286,88,194)(38,326,128,209,54,277,87,203)(39,317,127,218,55,268,86,171)(40,308,126,227,56,259,85,180)(41,299,125,236,57,250,84,189) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41),(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82),(83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123),(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164),(165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205),(206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246),(247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287),(288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328)], [(1,290,124,245,58,282,83,198),(2,322,164,213,59,273,123,166),(3,313,163,222,60,264,122,175),(4,304,162,231,61,255,121,184),(5,295,161,240,62,287,120,193),(6,327,160,208,63,278,119,202),(7,318,159,217,64,269,118,170),(8,309,158,226,65,260,117,179),(9,300,157,235,66,251,116,188),(10,291,156,244,67,283,115,197),(11,323,155,212,68,274,114,165),(12,314,154,221,69,265,113,174),(13,305,153,230,70,256,112,183),(14,296,152,239,71,247,111,192),(15,328,151,207,72,279,110,201),(16,319,150,216,73,270,109,169),(17,310,149,225,74,261,108,178),(18,301,148,234,75,252,107,187),(19,292,147,243,76,284,106,196),(20,324,146,211,77,275,105,205),(21,315,145,220,78,266,104,173),(22,306,144,229,79,257,103,182),(23,297,143,238,80,248,102,191),(24,288,142,206,81,280,101,200),(25,320,141,215,82,271,100,168),(26,311,140,224,42,262,99,177),(27,302,139,233,43,253,98,186),(28,293,138,242,44,285,97,195),(29,325,137,210,45,276,96,204),(30,316,136,219,46,267,95,172),(31,307,135,228,47,258,94,181),(32,298,134,237,48,249,93,190),(33,289,133,246,49,281,92,199),(34,321,132,214,50,272,91,167),(35,312,131,223,51,263,90,176),(36,303,130,232,52,254,89,185),(37,294,129,241,53,286,88,194),(38,326,128,209,54,277,87,203),(39,317,127,218,55,268,86,171),(40,308,126,227,56,259,85,180),(41,299,125,236,57,250,84,189)]])
Matrix representation of C41⋊2C8 ►in GL5(𝔽2297)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2296 | 1 | 0 | 0 |
| 0 | 2296 | 0 | 1 | 0 |
| 0 | 2296 | 0 | 0 | 1 |
| 0 | 2054 | 1653 | 644 | 242 |
| 1324 | 0 | 0 | 0 | 0 |
| 0 | 755 | 1383 | 1542 | 2237 |
| 0 | 862 | 1646 | 731 | 1804 |
| 0 | 275 | 1507 | 252 | 2234 |
| 0 | 523 | 603 | 396 | 1941 |
G:=sub<GL(5,GF(2297))| [1,0,0,0,0,0,2296,2296,2296,2054,0,1,0,0,1653,0,0,1,0,644,0,0,0,1,242],[1324,0,0,0,0,0,755,862,275,523,0,1383,1646,1507,603,0,1542,731,252,396,0,2237,1804,2234,1941] >;
C41⋊2C8 in GAP, Magma, Sage, TeX
C_{41}\rtimes_2C_8 % in TeX
G:=Group("C41:2C8"); // GroupNames label
G:=SmallGroup(328,3);
// by ID
G=gap.SmallGroup(328,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-41,8,21,4099,2567]);
// Polycyclic
G:=Group<a,b|a^41=b^8=1,b*a*b^-1=a^9>;
// generators/relations
Export
Subgroup lattice of C41⋊2C8 in TeX
Character table of C41⋊2C8 in TeX