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G = C2×C41⋊C4order 328 = 23·41

Direct product of C2 and C41⋊C4

Aliases: C2×C41⋊C4, C82⋊C4, D41⋊C4, D82.C2, D41.C22, C41⋊(C2×C4), SmallGroup(328,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C41 — C2×C41⋊C4
 Chief series C1 — C41 — D41 — C41⋊C4 — C2×C41⋊C4
 Lower central C41 — C2×C41⋊C4
 Upper central C1 — C2

Generators and relations for C2×C41⋊C4
G = < a,b,c | a2=b41=c4=1, ab=ba, ac=ca, cbc-1=b9 >

Character table of C2×C41⋊C4

 class 1 2A 2B 2C 4A 4B 4C 4D 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 -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 ρ4 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 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 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 ζ4140+ζ4132+ζ419+ζ41 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 orthogonal lifted from C41⋊C4 ρ10 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 -ζ4137-ζ4136-ζ415-ζ414 -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 orthogonal faithful ρ11 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 ζ4138+ζ4127+ζ4114+ζ413 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 orthogonal lifted from C41⋊C4 ρ12 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 -ζ4138-ζ4127-ζ4114-ζ413 -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 orthogonal faithful ρ13 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 -ζ4139-ζ4123-ζ4118-ζ412 -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 orthogonal faithful ρ14 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 -ζ4129-ζ4126-ζ4115-ζ4112 -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 orthogonal faithful ρ15 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 -ζ4130-ζ4124-ζ4117-ζ4111 -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 orthogonal faithful ρ16 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 -ζ4134-ζ4122-ζ4119-ζ417 -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 orthogonal faithful ρ17 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 ζ4139+ζ4123+ζ4118+ζ412 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 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 -ζ4133-ζ4131-ζ4110-ζ418 -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 orthogonal faithful ρ19 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 ζ4129+ζ4126+ζ4115+ζ4112 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 orthogonal lifted from C41⋊C4 ρ20 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 ζ4137+ζ4136+ζ415+ζ414 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 orthogonal lifted from C41⋊C4 ρ21 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 ζ4135+ζ4128+ζ4113+ζ416 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 orthogonal lifted from C41⋊C4 ρ22 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 -ζ4135-ζ4128-ζ4113-ζ416 -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 orthogonal faithful ρ23 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 -ζ4125-ζ4121-ζ4120-ζ4116 -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 orthogonal faithful ρ24 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 -ζ4140-ζ4132-ζ419-ζ41 -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 orthogonal faithful ρ25 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 ζ4125+ζ4121+ζ4120+ζ4116 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 orthogonal lifted from C41⋊C4 ρ26 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 ζ4133+ζ4131+ζ4110+ζ418 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 orthogonal lifted from C41⋊C4 ρ27 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 ζ4134+ζ4122+ζ4119+ζ417 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 orthogonal lifted from C41⋊C4 ρ28 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 ζ4130+ζ4124+ζ4117+ζ4111 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 orthogonal lifted from C41⋊C4

Smallest permutation representation of C2×C41⋊C4
On 82 points
Generators in S82
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)
(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)
(2 33 41 10)(3 24 40 19)(4 15 39 28)(5 6 38 37)(7 29 36 14)(8 20 35 23)(9 11 34 32)(12 25 31 18)(13 16 30 27)(17 21 26 22)(43 74 82 51)(44 65 81 60)(45 56 80 69)(46 47 79 78)(48 70 77 55)(49 61 76 64)(50 52 75 73)(53 66 72 59)(54 57 71 68)(58 62 67 63)

G:=sub<Sym(82)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82), (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), (2,33,41,10)(3,24,40,19)(4,15,39,28)(5,6,38,37)(7,29,36,14)(8,20,35,23)(9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)(43,74,82,51)(44,65,81,60)(45,56,80,69)(46,47,79,78)(48,70,77,55)(49,61,76,64)(50,52,75,73)(53,66,72,59)(54,57,71,68)(58,62,67,63)>;

G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82), (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), (2,33,41,10)(3,24,40,19)(4,15,39,28)(5,6,38,37)(7,29,36,14)(8,20,35,23)(9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)(43,74,82,51)(44,65,81,60)(45,56,80,69)(46,47,79,78)(48,70,77,55)(49,61,76,64)(50,52,75,73)(53,66,72,59)(54,57,71,68)(58,62,67,63) );

G=PermutationGroup([(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82)], [(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)], [(2,33,41,10),(3,24,40,19),(4,15,39,28),(5,6,38,37),(7,29,36,14),(8,20,35,23),(9,11,34,32),(12,25,31,18),(13,16,30,27),(17,21,26,22),(43,74,82,51),(44,65,81,60),(45,56,80,69),(46,47,79,78),(48,70,77,55),(49,61,76,64),(50,52,75,73),(53,66,72,59),(54,57,71,68),(58,62,67,63)])

Matrix representation of C2×C41⋊C4 in GL5(𝔽821)

 820 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 206 609 80 820 0 466 741 340 201 0 506 373 61 420 0 340 224 691 94
,
 295 0 0 0 0 0 725 146 284 161 0 685 489 288 706 0 735 35 225 599 0 320 772 668 203

G:=sub<GL(5,GF(821))| [820,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,206,466,506,340,0,609,741,373,224,0,80,340,61,691,0,820,201,420,94],[295,0,0,0,0,0,725,685,735,320,0,146,489,35,772,0,284,288,225,668,0,161,706,599,203] >;

C2×C41⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{41}\rtimes C_4
% in TeX

G:=Group("C2xC41:C4");
// GroupNames label

G:=SmallGroup(328,13);
// by ID

G=gap.SmallGroup(328,13);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,16,4099,1291]);
// Polycyclic

G:=Group<a,b,c|a^2=b^41=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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