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## G = C89⋊C4order 356 = 22·89

### The semidirect product of C89 and C4 acting faithfully

Aliases: C89⋊C4, D89.C2, SmallGroup(356,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C89 — C89⋊C4
 Chief series C1 — C89 — D89 — C89⋊C4
 Lower central C89 — C89⋊C4
 Upper central C1

Generators and relations for C89⋊C4
G = < a,b | a89=b4=1, bab-1=a55 >

Character table of C89⋊C4

 class 1 2 4A 4B 89A 89B 89C 89D 89E 89F 89G 89H 89I 89J 89K 89L 89M 89N 89O 89P 89Q 89R 89S 89T 89U 89V size 1 89 89 89 4 4 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 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 linear of order 2 ρ3 1 -1 -i i 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 4 ρ4 1 -1 i -i 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 4 ρ5 4 0 0 0 ζ8969+ζ8957+ζ8932+ζ8920 ζ8964+ζ8949+ζ8940+ζ8925 ζ8959+ζ8948+ζ8941+ζ8930 ζ8956+ζ8954+ζ8935+ζ8933 ζ8972+ζ8945+ζ8944+ζ8917 ζ8980+ζ8950+ζ8939+ζ899 ζ8988+ζ8955+ζ8934+ζ89 ζ8982+ζ8960+ζ8929+ζ897 ζ8970+ζ8966+ζ8923+ζ8919 ζ8975+ζ8958+ζ8931+ζ8914 ζ8985+ζ8947+ζ8942+ζ894 ζ8983+ζ8963+ζ8926+ζ896 ζ8978+ζ8971+ζ8918+ζ8911 ζ8987+ζ8968+ζ8921+ζ892 ζ8967+ζ8953+ζ8936+ζ8922 ζ8951+ζ8946+ζ8943+ζ8938 ζ8962+ζ8961+ζ8928+ζ8927 ζ8986+ζ8976+ζ8913+ζ893 ζ8977+ζ8952+ζ8937+ζ8912 ζ8984+ζ8981+ζ898+ζ895 ζ8979+ζ8973+ζ8916+ζ8910 ζ8974+ζ8965+ζ8924+ζ8915 orthogonal faithful ρ6 4 0 0 0 ζ8977+ζ8952+ζ8937+ζ8912 ζ8974+ζ8965+ζ8924+ζ8915 ζ8978+ζ8971+ζ8918+ζ8911 ζ8987+ζ8968+ζ8921+ζ892 ζ8962+ζ8961+ζ8928+ζ8927 ζ8959+ζ8948+ζ8941+ζ8930 ζ8956+ζ8954+ζ8935+ζ8933 ζ8967+ζ8953+ζ8936+ζ8922 ζ8985+ζ8947+ζ8942+ζ894 ζ8972+ζ8945+ζ8944+ζ8917 ζ8951+ζ8946+ζ8943+ζ8938 ζ8969+ζ8957+ζ8932+ζ8920 ζ8982+ζ8960+ζ8929+ζ897 ζ8970+ζ8966+ζ8923+ζ8919 ζ8975+ζ8958+ζ8931+ζ8914 ζ8984+ζ8981+ζ898+ζ895 ζ8988+ζ8955+ζ8934+ζ89 ζ8979+ζ8973+ζ8916+ζ8910 ζ8964+ζ8949+ζ8940+ζ8925 ζ8986+ζ8976+ζ8913+ζ893 ζ8983+ζ8963+ζ8926+ζ896 ζ8980+ζ8950+ζ8939+ζ899 orthogonal faithful ρ7 4 0 0 0 ζ8962+ζ8961+ζ8928+ζ8927 ζ8956+ζ8954+ζ8935+ζ8933 ζ8985+ζ8947+ζ8942+ζ894 ζ8964+ζ8949+ζ8940+ζ8925 ζ8983+ζ8963+ζ8926+ζ896 ζ8970+ζ8966+ζ8923+ζ8919 ζ8977+ζ8952+ζ8937+ζ8912 ζ8984+ζ8981+ζ898+ζ895 ζ8980+ζ8950+ζ8939+ζ899 ζ8979+ζ8973+ζ8916+ζ8910 ζ8959+ζ8948+ζ8941+ζ8930 ζ8972+ζ8945+ζ8944+ζ8917 ζ8951+ζ8946+ζ8943+ζ8938 ζ8974+ζ8965+ζ8924+ζ8915 ζ8986+ζ8976+ζ8913+ζ893 ζ8978+ζ8971+ζ8918+ζ8911 ζ8969+ζ8957+ζ8932+ζ8920 ζ8967+ζ8953+ζ8936+ζ8922 ζ8988+ζ8955+ζ8934+ζ89 ζ8982+ζ8960+ζ8929+ζ897 ζ8975+ζ8958+ζ8931+ζ8914 ζ8987+ζ8968+ζ8921+ζ892 orthogonal faithful ρ8 4 0 0 0 ζ8982+ζ8960+ζ8929+ζ897 ζ8975+ζ8958+ζ8931+ζ8914 ζ8988+ζ8955+ζ8934+ζ89 ζ8979+ζ8973+ζ8916+ζ8910 ζ8951+ζ8946+ζ8943+ζ8938 ζ8962+ζ8961+ζ8928+ζ8927 ζ8986+ζ8976+ζ8913+ζ893 ζ8987+ζ8968+ζ8921+ζ892 ζ8969+ζ8957+ζ8932+ζ8920 ζ8985+ζ8947+ζ8942+ζ894 ζ8977+ζ8952+ζ8937+ζ8912 ζ8978+ζ8971+ζ8918+ζ8911 ζ8956+ζ8954+ζ8935+ζ8933 ζ8983+ζ8963+ζ8926+ζ896 ζ8970+ζ8966+ζ8923+ζ8919 ζ8964+ζ8949+ζ8940+ζ8925 ζ8984+ζ8981+ζ898+ζ895 ζ8980+ζ8950+ζ8939+ζ899 ζ8967+ζ8953+ζ8936+ζ8922 ζ8974+ζ8965+ζ8924+ζ8915 ζ8959+ζ8948+ζ8941+ζ8930 ζ8972+ζ8945+ζ8944+ζ8917 orthogonal faithful ρ9 4 0 0 0 ζ8974+ζ8965+ζ8924+ζ8915 ζ8959+ζ8948+ζ8941+ζ8930 ζ8967+ζ8953+ζ8936+ζ8922 ζ8985+ζ8947+ζ8942+ζ894 ζ8956+ζ8954+ζ8935+ζ8933 ζ8982+ζ8960+ζ8929+ζ897 ζ8970+ζ8966+ζ8923+ζ8919 ζ8972+ζ8945+ζ8944+ζ8917 ζ8984+ζ8981+ζ898+ζ895 ζ8988+ζ8955+ζ8934+ζ89 ζ8986+ζ8976+ζ8913+ζ893 ζ8964+ζ8949+ζ8940+ζ8925 ζ8975+ζ8958+ζ8931+ζ8914 ζ8951+ζ8946+ζ8943+ζ8938 ζ8962+ζ8961+ζ8928+ζ8927 ζ8979+ζ8973+ζ8916+ζ8910 ζ8987+ζ8968+ζ8921+ζ892 ζ8969+ζ8957+ζ8932+ζ8920 ζ8980+ζ8950+ζ8939+ζ899 ζ8983+ζ8963+ζ8926+ζ896 ζ8977+ζ8952+ζ8937+ζ8912 ζ8978+ζ8971+ζ8918+ζ8911 orthogonal faithful ρ10 4 0 0 0 ζ8959+ζ8948+ζ8941+ζ8930 ζ8982+ζ8960+ζ8929+ζ897 ζ8972+ζ8945+ζ8944+ζ8917 ζ8984+ζ8981+ζ898+ζ895 ζ8970+ζ8966+ζ8923+ζ8919 ζ8975+ζ8958+ζ8931+ζ8914 ζ8951+ζ8946+ζ8943+ζ8938 ζ8988+ζ8955+ζ8934+ζ89 ζ8979+ζ8973+ζ8916+ζ8910 ζ8987+ζ8968+ζ8921+ζ892 ζ8983+ζ8963+ζ8926+ζ896 ζ8980+ζ8950+ζ8939+ζ899 ζ8962+ζ8961+ζ8928+ζ8927 ζ8986+ζ8976+ζ8913+ζ893 ζ8956+ζ8954+ζ8935+ζ8933 ζ8969+ζ8957+ζ8932+ζ8920 ζ8985+ζ8947+ζ8942+ζ894 ζ8964+ζ8949+ζ8940+ζ8925 ζ8978+ζ8971+ζ8918+ζ8911 ζ8977+ζ8952+ζ8937+ζ8912 ζ8974+ζ8965+ζ8924+ζ8915 ζ8967+ζ8953+ζ8936+ζ8922 orthogonal faithful ρ11 4 0 0 0 ζ8951+ζ8946+ζ8943+ζ8938 ζ8986+ζ8976+ζ8913+ζ893 ζ8969+ζ8957+ζ8932+ζ8920 ζ8967+ζ8953+ζ8936+ζ8922 ζ8959+ζ8948+ζ8941+ζ8930 ζ8983+ζ8963+ζ8926+ζ896 ζ8982+ζ8960+ζ8929+ζ897 ζ8964+ζ8949+ζ8940+ζ8925 ζ8972+ζ8945+ζ8944+ζ8917 ζ8980+ζ8950+ζ8939+ζ899 ζ8962+ζ8961+ζ8928+ζ8927 ζ8985+ζ8947+ζ8942+ζ894 ζ8977+ζ8952+ζ8937+ζ8912 ζ8975+ζ8958+ζ8931+ζ8914 ζ8974+ζ8965+ζ8924+ζ8915 ζ8988+ζ8955+ζ8934+ζ89 ζ8978+ζ8971+ζ8918+ζ8911 ζ8987+ζ8968+ζ8921+ζ892 ζ8984+ζ8981+ζ898+ζ895 ζ8956+ζ8954+ζ8935+ζ8933 ζ8970+ζ8966+ζ8923+ζ8919 ζ8979+ζ8973+ζ8916+ζ8910 orthogonal faithful ρ12 4 0 0 0 ζ8967+ζ8953+ζ8936+ζ8922 ζ8972+ζ8945+ζ8944+ζ8917 ζ8956+ζ8954+ζ8935+ζ8933 ζ8983+ζ8963+ζ8926+ζ896 ζ8984+ζ8981+ζ898+ζ895 ζ8988+ζ8955+ζ8934+ζ89 ζ8979+ζ8973+ζ8916+ζ8910 ζ8970+ζ8966+ζ8923+ζ8919 ζ8977+ζ8952+ζ8937+ζ8912 ζ8951+ζ8946+ζ8943+ζ8938 ζ8964+ζ8949+ζ8940+ζ8925 ζ8982+ζ8960+ζ8929+ζ897 ζ8987+ζ8968+ζ8921+ζ892 ζ8969+ζ8957+ζ8932+ζ8920 ζ8985+ζ8947+ζ8942+ζ894 ζ8974+ζ8965+ζ8924+ζ8915 ζ8986+ζ8976+ζ8913+ζ893 ζ8959+ζ8948+ζ8941+ζ8930 ζ8975+ζ8958+ζ8931+ζ8914 ζ8980+ζ8950+ζ8939+ζ899 ζ8978+ζ8971+ζ8918+ζ8911 ζ8962+ζ8961+ζ8928+ζ8927 orthogonal faithful ρ13 4 0 0 0 ζ8970+ζ8966+ζ8923+ζ8919 ζ8951+ζ8946+ζ8943+ζ8938 ζ8979+ζ8973+ζ8916+ζ8910 ζ8978+ζ8971+ζ8918+ζ8911 ζ8974+ζ8965+ζ8924+ζ8915 ζ8986+ζ8976+ζ8913+ζ893 ζ8959+ζ8948+ζ8941+ζ8930 ζ8969+ζ8957+ζ8932+ζ8920 ζ8967+ζ8953+ζ8936+ζ8922 ζ8964+ζ8949+ζ8940+ζ8925 ζ8975+ζ8958+ζ8931+ζ8914 ζ8987+ζ8968+ζ8921+ζ892 ζ8983+ζ8963+ζ8926+ζ896 ζ8982+ζ8960+ζ8929+ζ897 ζ8977+ζ8952+ζ8937+ζ8912 ζ8972+ζ8945+ζ8944+ζ8917 ζ8980+ζ8950+ζ8939+ζ899 ζ8988+ζ8955+ζ8934+ζ89 ζ8985+ζ8947+ζ8942+ζ894 ζ8962+ζ8961+ζ8928+ζ8927 ζ8956+ζ8954+ζ8935+ζ8933 ζ8984+ζ8981+ζ898+ζ895 orthogonal faithful ρ14 4 0 0 0 ζ8980+ζ8950+ζ8939+ζ899 ζ8978+ζ8971+ζ8918+ζ8911 ζ8975+ζ8958+ζ8931+ζ8914 ζ8951+ζ8946+ζ8943+ζ8938 ζ8987+ζ8968+ζ8921+ζ892 ζ8967+ζ8953+ζ8936+ζ8922 ζ8985+ζ8947+ζ8942+ζ894 ζ8962+ζ8961+ζ8928+ζ8927 ζ8986+ζ8976+ζ8913+ζ893 ζ8956+ζ8954+ζ8935+ζ8933 ζ8979+ζ8973+ζ8916+ζ8910 ζ8974+ζ8965+ζ8924+ζ8915 ζ8972+ζ8945+ζ8944+ζ8917 ζ8984+ζ8981+ζ898+ζ895 ζ8988+ζ8955+ζ8934+ζ89 ζ8983+ζ8963+ζ8926+ζ896 ζ8970+ζ8966+ζ8923+ζ8919 ζ8977+ζ8952+ζ8937+ζ8912 ζ8959+ζ8948+ζ8941+ζ8930 ζ8969+ζ8957+ζ8932+ζ8920 ζ8964+ζ8949+ζ8940+ζ8925 ζ8982+ζ8960+ζ8929+ζ897 orthogonal faithful ρ15 4 0 0 0 ζ8975+ζ8958+ζ8931+ζ8914 ζ8962+ζ8961+ζ8928+ζ8927 ζ8987+ζ8968+ζ8921+ζ892 ζ8969+ζ8957+ζ8932+ζ8920 ζ8986+ζ8976+ζ8913+ζ893 ζ8956+ζ8954+ζ8935+ζ8933 ζ8983+ζ8963+ζ8926+ζ896 ζ8985+ζ8947+ζ8942+ζ894 ζ8964+ζ8949+ζ8940+ζ8925 ζ8984+ζ8981+ζ898+ζ895 ζ8974+ζ8965+ζ8924+ζ8915 ζ8967+ζ8953+ζ8936+ζ8922 ζ8970+ζ8966+ζ8923+ζ8919 ζ8977+ζ8952+ζ8937+ζ8912 ζ8951+ζ8946+ζ8943+ζ8938 ζ8980+ζ8950+ζ8939+ζ899 ζ8979+ζ8973+ζ8916+ζ8910 ζ8978+ζ8971+ζ8918+ζ8911 ζ8972+ζ8945+ζ8944+ζ8917 ζ8959+ζ8948+ζ8941+ζ8930 ζ8982+ζ8960+ζ8929+ζ897 ζ8988+ζ8955+ζ8934+ζ89 orthogonal faithful ρ16 4 0 0 0 ζ8988+ζ8955+ζ8934+ζ89 ζ8987+ζ8968+ζ8921+ζ892 ζ8951+ζ8946+ζ8943+ζ8938 ζ8974+ζ8965+ζ8924+ζ8915 ζ8969+ζ8957+ζ8932+ζ8920 ζ8985+ζ8947+ζ8942+ζ894 ζ8964+ζ8949+ζ8940+ζ8925 ζ8986+ζ8976+ζ8913+ζ893 ζ8959+ζ8948+ζ8941+ζ8930 ζ8983+ζ8963+ζ8926+ζ896 ζ8978+ζ8971+ζ8918+ζ8911 ζ8962+ζ8961+ζ8928+ζ8927 ζ8984+ζ8981+ζ898+ζ895 ζ8980+ζ8950+ζ8939+ζ899 ζ8979+ζ8973+ζ8916+ζ8910 ζ8982+ζ8960+ζ8929+ζ897 ζ8977+ζ8952+ζ8937+ζ8912 ζ8975+ζ8958+ζ8931+ζ8914 ζ8956+ζ8954+ζ8935+ζ8933 ζ8967+ζ8953+ζ8936+ζ8922 ζ8972+ζ8945+ζ8944+ζ8917 ζ8970+ζ8966+ζ8923+ζ8919 orthogonal faithful ρ17 4 0 0 0 ζ8956+ζ8954+ζ8935+ζ8933 ζ8970+ζ8966+ζ8923+ζ8919 ζ8984+ζ8981+ζ898+ζ895 ζ8980+ζ8950+ζ8939+ζ899 ζ8977+ζ8952+ζ8937+ζ8912 ζ8951+ζ8946+ζ8943+ζ8938 ζ8974+ζ8965+ζ8924+ζ8915 ζ8979+ζ8973+ζ8916+ζ8910 ζ8978+ζ8971+ζ8918+ζ8911 ζ8969+ζ8957+ζ8932+ζ8920 ζ8982+ζ8960+ζ8929+ζ897 ζ8988+ζ8955+ζ8934+ζ89 ζ8986+ζ8976+ζ8913+ζ893 ζ8959+ζ8948+ζ8941+ζ8930 ζ8983+ζ8963+ζ8926+ζ896 ζ8967+ζ8953+ζ8936+ζ8922 ζ8964+ζ8949+ζ8940+ζ8925 ζ8972+ζ8945+ζ8944+ζ8917 ζ8987+ζ8968+ζ8921+ζ892 ζ8975+ζ8958+ζ8931+ζ8914 ζ8962+ζ8961+ζ8928+ζ8927 ζ8985+ζ8947+ζ8942+ζ894 orthogonal faithful ρ18 4 0 0 0 ζ8986+ζ8976+ζ8913+ζ893 ζ8983+ζ8963+ζ8926+ζ896 ζ8964+ζ8949+ζ8940+ζ8925 ζ8972+ζ8945+ζ8944+ζ8917 ζ8982+ζ8960+ζ8929+ζ897 ζ8977+ζ8952+ζ8937+ζ8912 ζ8975+ζ8958+ζ8931+ζ8914 ζ8980+ζ8950+ζ8939+ζ899 ζ8988+ζ8955+ζ8934+ζ89 ζ8978+ζ8971+ζ8918+ζ8911 ζ8956+ζ8954+ζ8935+ζ8933 ζ8984+ζ8981+ζ898+ζ895 ζ8974+ζ8965+ζ8924+ζ8915 ζ8962+ζ8961+ζ8928+ζ8927 ζ8959+ζ8948+ζ8941+ζ8930 ζ8987+ζ8968+ζ8921+ζ892 ζ8967+ζ8953+ζ8936+ζ8922 ζ8985+ζ8947+ζ8942+ζ894 ζ8979+ζ8973+ζ8916+ζ8910 ζ8970+ζ8966+ζ8923+ζ8919 ζ8951+ζ8946+ζ8943+ζ8938 ζ8969+ζ8957+ζ8932+ζ8920 orthogonal faithful ρ19 4 0 0 0 ζ8985+ζ8947+ζ8942+ζ894 ζ8984+ζ8981+ζ898+ζ895 ζ8983+ζ8963+ζ8926+ζ896 ζ8982+ζ8960+ζ8929+ζ897 ζ8980+ζ8950+ζ8939+ζ899 ζ8979+ζ8973+ζ8916+ζ8910 ζ8978+ζ8971+ζ8918+ζ8911 ζ8977+ζ8952+ζ8937+ζ8912 ζ8975+ζ8958+ζ8931+ζ8914 ζ8974+ζ8965+ζ8924+ζ8915 ζ8972+ζ8945+ζ8944+ζ8917 ζ8970+ζ8966+ζ8923+ζ8919 ζ8969+ζ8957+ζ8932+ζ8920 ζ8967+ζ8953+ζ8936+ζ8922 ζ8964+ζ8949+ζ8940+ζ8925 ζ8962+ζ8961+ζ8928+ζ8927 ζ8959+ζ8948+ζ8941+ζ8930 ζ8956+ζ8954+ζ8935+ζ8933 ζ8951+ζ8946+ζ8943+ζ8938 ζ8988+ζ8955+ζ8934+ζ89 ζ8987+ζ8968+ζ8921+ζ892 ζ8986+ζ8976+ζ8913+ζ893 orthogonal faithful ρ20 4 0 0 0 ζ8972+ζ8945+ζ8944+ζ8917 ζ8988+ζ8955+ζ8934+ζ89 ζ8970+ζ8966+ζ8923+ζ8919 ζ8977+ζ8952+ζ8937+ζ8912 ζ8979+ζ8973+ζ8916+ζ8910 ζ8987+ζ8968+ζ8921+ζ892 ζ8969+ζ8957+ζ8932+ζ8920 ζ8951+ζ8946+ζ8943+ζ8938 ζ8974+ζ8965+ζ8924+ζ8915 ζ8986+ζ8976+ζ8913+ζ893 ζ8980+ζ8950+ζ8939+ζ899 ζ8975+ζ8958+ζ8931+ζ8914 ζ8985+ζ8947+ζ8942+ζ894 ζ8964+ζ8949+ζ8940+ζ8925 ζ8984+ζ8981+ζ898+ζ895 ζ8959+ζ8948+ζ8941+ζ8930 ζ8983+ζ8963+ζ8926+ζ896 ζ8982+ζ8960+ζ8929+ζ897 ζ8962+ζ8961+ζ8928+ζ8927 ζ8978+ζ8971+ζ8918+ζ8911 ζ8967+ζ8953+ζ8936+ζ8922 ζ8956+ζ8954+ζ8935+ζ8933 orthogonal faithful ρ21 4 0 0 0 ζ8983+ζ8963+ζ8926+ζ896 ζ8977+ζ8952+ζ8937+ζ8912 ζ8980+ζ8950+ζ8939+ζ899 ζ8988+ζ8955+ζ8934+ζ89 ζ8975+ζ8958+ζ8931+ζ8914 ζ8974+ζ8965+ζ8924+ζ8915 ζ8962+ζ8961+ζ8928+ζ8927 ζ8978+ζ8971+ζ8918+ζ8911 ζ8987+ζ8968+ζ8921+ζ892 ζ8967+ζ8953+ζ8936+ζ8922 ζ8970+ζ8966+ζ8923+ζ8919 ζ8979+ζ8973+ζ8916+ζ8910 ζ8959+ζ8948+ζ8941+ζ8930 ζ8956+ζ8954+ζ8935+ζ8933 ζ8982+ζ8960+ζ8929+ζ897 ζ8985+ζ8947+ζ8942+ζ894 ζ8972+ζ8945+ζ8944+ζ8917 ζ8984+ζ8981+ζ898+ζ895 ζ8969+ζ8957+ζ8932+ζ8920 ζ8951+ζ8946+ζ8943+ζ8938 ζ8986+ζ8976+ζ8913+ζ893 ζ8964+ζ8949+ζ8940+ζ8925 orthogonal faithful ρ22 4 0 0 0 ζ8979+ζ8973+ζ8916+ζ8910 ζ8969+ζ8957+ζ8932+ζ8920 ζ8974+ζ8965+ζ8924+ζ8915 ζ8962+ζ8961+ζ8928+ζ8927 ζ8967+ζ8953+ζ8936+ζ8922 ζ8964+ζ8949+ζ8940+ζ8925 ζ8972+ζ8945+ζ8944+ζ8917 ζ8959+ζ8948+ζ8941+ζ8930 ζ8956+ζ8954+ζ8935+ζ8933 ζ8982+ζ8960+ζ8929+ζ897 ζ8987+ζ8968+ζ8921+ζ892 ζ8986+ζ8976+ζ8913+ζ893 ζ8980+ζ8950+ζ8939+ζ899 ζ8988+ζ8955+ζ8934+ζ89 ζ8978+ζ8971+ζ8918+ζ8911 ζ8970+ζ8966+ζ8923+ζ8919 ζ8975+ζ8958+ζ8931+ζ8914 ζ8951+ζ8946+ζ8943+ζ8938 ζ8983+ζ8963+ζ8926+ζ896 ζ8985+ζ8947+ζ8942+ζ894 ζ8984+ζ8981+ζ898+ζ895 ζ8977+ζ8952+ζ8937+ζ8912 orthogonal faithful ρ23 4 0 0 0 ζ8964+ζ8949+ζ8940+ζ8925 ζ8980+ζ8950+ζ8939+ζ899 ζ8982+ζ8960+ζ8929+ζ897 ζ8970+ζ8966+ζ8923+ζ8919 ζ8988+ζ8955+ζ8934+ζ89 ζ8978+ζ8971+ζ8918+ζ8911 ζ8987+ζ8968+ζ8921+ζ892 ζ8975+ζ8958+ζ8931+ζ8914 ζ8951+ζ8946+ζ8943+ζ8938 ζ8962+ζ8961+ζ8928+ζ8927 ζ8984+ζ8981+ζ898+ζ895 ζ8977+ζ8952+ζ8937+ζ8912 ζ8967+ζ8953+ζ8936+ζ8922 ζ8985+ζ8947+ζ8942+ζ894 ζ8972+ζ8945+ζ8944+ζ8917 ζ8986+ζ8976+ζ8913+ζ893 ζ8956+ζ8954+ζ8935+ζ8933 ζ8983+ζ8963+ζ8926+ζ896 ζ8974+ζ8965+ζ8924+ζ8915 ζ8979+ζ8973+ζ8916+ζ8910 ζ8969+ζ8957+ζ8932+ζ8920 ζ8959+ζ8948+ζ8941+ζ8930 orthogonal faithful ρ24 4 0 0 0 ζ8984+ζ8981+ζ898+ζ895 ζ8979+ζ8973+ζ8916+ζ8910 ζ8977+ζ8952+ζ8937+ζ8912 ζ8975+ζ8958+ζ8931+ζ8914 ζ8978+ζ8971+ζ8918+ζ8911 ζ8969+ζ8957+ζ8932+ζ8920 ζ8967+ζ8953+ζ8936+ζ8922 ζ8974+ζ8965+ζ8924+ζ8915 ζ8962+ζ8961+ζ8928+ζ8927 ζ8959+ζ8948+ζ8941+ζ8930 ζ8988+ζ8955+ζ8934+ζ89 ζ8951+ζ8946+ζ8943+ζ8938 ζ8964+ζ8949+ζ8940+ζ8925 ζ8972+ζ8945+ζ8944+ζ8917 ζ8980+ζ8950+ζ8939+ζ899 ζ8956+ζ8954+ζ8935+ζ8933 ζ8982+ζ8960+ζ8929+ζ897 ζ8970+ζ8966+ζ8923+ζ8919 ζ8986+ζ8976+ζ8913+ζ893 ζ8987+ζ8968+ζ8921+ζ892 ζ8985+ζ8947+ζ8942+ζ894 ζ8983+ζ8963+ζ8926+ζ896 orthogonal faithful ρ25 4 0 0 0 ζ8987+ζ8968+ζ8921+ζ892 ζ8985+ζ8947+ζ8942+ζ894 ζ8986+ζ8976+ζ8913+ζ893 ζ8959+ζ8948+ζ8941+ζ8930 ζ8964+ζ8949+ζ8940+ζ8925 ζ8984+ζ8981+ζ898+ζ895 ζ8980+ζ8950+ζ8939+ζ899 ζ8983+ζ8963+ζ8926+ζ896 ζ8982+ζ8960+ζ8929+ζ897 ζ8977+ζ8952+ζ8937+ζ8912 ζ8967+ζ8953+ζ8936+ζ8922 ζ8956+ζ8954+ζ8935+ζ8933 ζ8979+ζ8973+ζ8916+ζ8910 ζ8978+ζ8971+ζ8918+ζ8911 ζ8969+ζ8957+ζ8932+ζ8920 ζ8975+ζ8958+ζ8931+ζ8914 ζ8974+ζ8965+ζ8924+ζ8915 ζ8962+ζ8961+ζ8928+ζ8927 ζ8970+ζ8966+ζ8923+ζ8919 ζ8972+ζ8945+ζ8944+ζ8917 ζ8988+ζ8955+ζ8934+ζ89 ζ8951+ζ8946+ζ8943+ζ8938 orthogonal faithful ρ26 4 0 0 0 ζ8978+ζ8971+ζ8918+ζ8911 ζ8967+ζ8953+ζ8936+ζ8922 ζ8962+ζ8961+ζ8928+ζ8927 ζ8986+ζ8976+ζ8913+ζ893 ζ8985+ζ8947+ζ8942+ζ894 ζ8972+ζ8945+ζ8944+ζ8917 ζ8984+ζ8981+ζ898+ζ895 ζ8956+ζ8954+ζ8935+ζ8933 ζ8983+ζ8963+ζ8926+ζ896 ζ8970+ζ8966+ζ8923+ζ8919 ζ8969+ζ8957+ζ8932+ζ8920 ζ8959+ζ8948+ζ8941+ζ8930 ζ8988+ζ8955+ζ8934+ζ89 ζ8979+ζ8973+ζ8916+ζ8910 ζ8987+ζ8968+ζ8921+ζ892 ζ8977+ζ8952+ζ8937+ζ8912 ζ8951+ζ8946+ζ8943+ζ8938 ζ8974+ζ8965+ζ8924+ζ8915 ζ8982+ζ8960+ζ8929+ζ897 ζ8964+ζ8949+ζ8940+ζ8925 ζ8980+ζ8950+ζ8939+ζ899 ζ8975+ζ8958+ζ8931+ζ8914 orthogonal faithful

Smallest permutation representation of C89⋊C4
On 89 points: primitive
Generators in S89
```(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)
(2 35 89 56)(3 69 88 22)(4 14 87 77)(5 48 86 43)(6 82 85 9)(7 27 84 64)(8 61 83 30)(10 40 81 51)(11 74 80 17)(12 19 79 72)(13 53 78 38)(15 32 76 59)(16 66 75 25)(18 45 73 46)(20 24 71 67)(21 58 70 33)(23 37 68 54)(26 50 65 41)(28 29 63 62)(31 42 60 49)(34 55 57 36)(39 47 52 44)```

`G:=sub<Sym(89)| (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), (2,35,89,56)(3,69,88,22)(4,14,87,77)(5,48,86,43)(6,82,85,9)(7,27,84,64)(8,61,83,30)(10,40,81,51)(11,74,80,17)(12,19,79,72)(13,53,78,38)(15,32,76,59)(16,66,75,25)(18,45,73,46)(20,24,71,67)(21,58,70,33)(23,37,68,54)(26,50,65,41)(28,29,63,62)(31,42,60,49)(34,55,57,36)(39,47,52,44)>;`

`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), (2,35,89,56)(3,69,88,22)(4,14,87,77)(5,48,86,43)(6,82,85,9)(7,27,84,64)(8,61,83,30)(10,40,81,51)(11,74,80,17)(12,19,79,72)(13,53,78,38)(15,32,76,59)(16,66,75,25)(18,45,73,46)(20,24,71,67)(21,58,70,33)(23,37,68,54)(26,50,65,41)(28,29,63,62)(31,42,60,49)(34,55,57,36)(39,47,52,44) );`

`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)], [(2,35,89,56),(3,69,88,22),(4,14,87,77),(5,48,86,43),(6,82,85,9),(7,27,84,64),(8,61,83,30),(10,40,81,51),(11,74,80,17),(12,19,79,72),(13,53,78,38),(15,32,76,59),(16,66,75,25),(18,45,73,46),(20,24,71,67),(21,58,70,33),(23,37,68,54),(26,50,65,41),(28,29,63,62),(31,42,60,49),(34,55,57,36),(39,47,52,44)]])`

Matrix representation of C89⋊C4 in GL4(𝔽1069) generated by

 524 1 0 0 311 0 1 0 596 0 0 1 462 625 629 1053
,
 781 1028 1028 875 183 481 245 320 781 838 824 612 634 261 462 52
`G:=sub<GL(4,GF(1069))| [524,311,596,462,1,0,0,625,0,1,0,629,0,0,1,1053],[781,183,781,634,1028,481,838,261,1028,245,824,462,875,320,612,52] >;`

C89⋊C4 in GAP, Magma, Sage, TeX

`C_{89}\rtimes C_4`
`% in TeX`

`G:=Group("C89:C4");`
`// GroupNames label`

`G:=SmallGroup(356,3);`
`// by ID`

`G=gap.SmallGroup(356,3);`
`# by ID`

`G:=PCGroup([3,-2,-2,-89,6,1226,1589]);`
`// Polycyclic`

`G:=Group<a,b|a^89=b^4=1,b*a*b^-1=a^55>;`
`// generators/relations`

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