Copied to
clipboard

G = C73⋊C4  order 292 = 22·73

The semidirect product of C73 and C4 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C73⋊C4, D73.C2, SmallGroup(292,3)

Series: Derived ►Chief ►Lower central ►Upper central

C1 — C73 — C73⋊C4
C1 — C73 — D73 — C73⋊C4
C73 — C73⋊C4
C1

Generators and relations for C73⋊C4
 G = < a,b | a73=b4=1, bab-1=a46 >

73C2
73C4

Character table of C73⋊C4

 class 124A4B73A73B73C73D73E73F73G73H73I73J73K73L73M73N73O73P73Q73R
 size 1737373444444444444444444
ρ11111111111111111111111    trivial
ρ211-1-1111111111111111111    linear of order 2
ρ31-1i-i111111111111111111    linear of order 4
ρ41-1-ii111111111111111111    linear of order 4
ρ54000ζ7369+ζ7338+ζ7335+ζ734ζ7368+ζ7362+ζ7311+ζ735ζ7367+ζ7357+ζ7316+ζ736ζ7366+ζ7343+ζ7330+ζ737ζ7364+ζ7349+ζ7324+ζ739ζ7363+ζ7351+ζ7322+ζ7310ζ7361+ζ7341+ζ7332+ζ7312ζ7360+ζ7359+ζ7314+ζ7313ζ7358+ζ7340+ζ7333+ζ7315ζ7356+ζ7352+ζ7321+ζ7317ζ7355+ζ7348+ζ7325+ζ7318ζ7353+ζ7344+ζ7329+ζ7320ζ7350+ζ7337+ζ7336+ζ7323ζ7347+ζ7345+ζ7328+ζ7326ζ7342+ζ7339+ζ7334+ζ7331ζ7372+ζ7346+ζ7327+ζ73ζ7371+ζ7354+ζ7319+ζ732ζ7370+ζ7365+ζ738+ζ733    orthogonal faithful
ρ64000ζ7350+ζ7337+ζ7336+ζ7323ζ7347+ζ7345+ζ7328+ζ7326ζ7371+ζ7354+ζ7319+ζ732ζ7363+ζ7351+ζ7322+ζ7310ζ7370+ζ7365+ζ738+ζ733ζ7356+ζ7352+ζ7321+ζ7317ζ7369+ζ7338+ζ7335+ζ734ζ7353+ζ7344+ζ7329+ζ7320ζ7368+ζ7362+ζ7311+ζ735ζ7366+ζ7343+ζ7330+ζ737ζ7367+ζ7357+ζ7316+ζ736ζ7342+ζ7339+ζ7334+ζ7331ζ7361+ζ7341+ζ7332+ζ7312ζ7358+ζ7340+ζ7333+ζ7315ζ7360+ζ7359+ζ7314+ζ7313ζ7364+ζ7349+ζ7324+ζ739ζ7355+ζ7348+ζ7325+ζ7318ζ7372+ζ7346+ζ7327+ζ73    orthogonal faithful
ρ74000ζ7356+ζ7352+ζ7321+ζ7317ζ7370+ζ7365+ζ738+ζ733ζ7368+ζ7362+ζ7311+ζ735ζ7355+ζ7348+ζ7325+ζ7318ζ7353+ζ7344+ζ7329+ζ7320ζ7367+ζ7357+ζ7316+ζ736ζ7363+ζ7351+ζ7322+ζ7310ζ7350+ζ7337+ζ7336+ζ7323ζ7364+ζ7349+ζ7324+ζ739ζ7371+ζ7354+ζ7319+ζ732ζ7358+ζ7340+ζ7333+ζ7315ζ7361+ζ7341+ζ7332+ζ7312ζ7366+ζ7343+ζ7330+ζ737ζ7372+ζ7346+ζ7327+ζ73ζ7369+ζ7338+ζ7335+ζ734ζ7360+ζ7359+ζ7314+ζ7313ζ7347+ζ7345+ζ7328+ζ7326ζ7342+ζ7339+ζ7334+ζ7331    orthogonal faithful
ρ84000ζ7371+ζ7354+ζ7319+ζ732ζ7342+ζ7339+ζ7334+ζ7331ζ7370+ζ7365+ζ738+ζ733ζ7358+ζ7340+ζ7333+ζ7315ζ7361+ζ7341+ζ7332+ζ7312ζ7368+ζ7362+ζ7311+ζ735ζ7367+ζ7357+ζ7316+ζ736ζ7366+ζ7343+ζ7330+ζ737ζ7353+ζ7344+ζ7329+ζ7320ζ7347+ζ7345+ζ7328+ζ7326ζ7364+ζ7349+ζ7324+ζ739ζ7363+ζ7351+ζ7322+ζ7310ζ7355+ζ7348+ζ7325+ζ7318ζ7360+ζ7359+ζ7314+ζ7313ζ7356+ζ7352+ζ7321+ζ7317ζ7350+ζ7337+ζ7336+ζ7323ζ7372+ζ7346+ζ7327+ζ73ζ7369+ζ7338+ζ7335+ζ734    orthogonal faithful
ρ94000ζ7360+ζ7359+ζ7314+ζ7313ζ7371+ζ7354+ζ7319+ζ732ζ7356+ζ7352+ζ7321+ζ7317ζ7361+ζ7341+ζ7332+ζ7312ζ7368+ζ7362+ζ7311+ζ735ζ7369+ζ7338+ζ7335+ζ734ζ7342+ζ7339+ζ7334+ζ7331ζ7364+ζ7349+ζ7324+ζ739ζ7367+ζ7357+ζ7316+ζ736ζ7350+ζ7337+ζ7336+ζ7323ζ7363+ζ7351+ζ7322+ζ7310ζ7370+ζ7365+ζ738+ζ733ζ7353+ζ7344+ζ7329+ζ7320ζ7355+ζ7348+ζ7325+ζ7318ζ7372+ζ7346+ζ7327+ζ73ζ7358+ζ7340+ζ7333+ζ7315ζ7366+ζ7343+ζ7330+ζ737ζ7347+ζ7345+ζ7328+ζ7326    orthogonal faithful
ρ104000ζ7363+ζ7351+ζ7322+ζ7310ζ7364+ζ7349+ζ7324+ζ739ζ7358+ζ7340+ζ7333+ζ7315ζ7371+ζ7354+ζ7319+ζ732ζ7360+ζ7359+ζ7314+ζ7313ζ7355+ζ7348+ζ7325+ζ7318ζ7366+ζ7343+ζ7330+ζ737ζ7369+ζ7338+ζ7335+ζ734ζ7372+ζ7346+ζ7327+ζ73ζ7367+ζ7357+ζ7316+ζ736ζ7347+ζ7345+ζ7328+ζ7326ζ7350+ζ7337+ζ7336+ζ7323ζ7356+ζ7352+ζ7321+ζ7317ζ7370+ζ7365+ζ738+ζ733ζ7361+ζ7341+ζ7332+ζ7312ζ7342+ζ7339+ζ7334+ζ7331ζ7368+ζ7362+ζ7311+ζ735ζ7353+ζ7344+ζ7329+ζ7320    orthogonal faithful
ρ114000ζ7353+ζ7344+ζ7329+ζ7320ζ7355+ζ7348+ζ7325+ζ7318ζ7366+ζ7343+ζ7330+ζ737ζ7369+ζ7338+ζ7335+ζ734ζ7347+ζ7345+ζ7328+ζ7326ζ7350+ζ7337+ζ7336+ζ7323ζ7360+ζ7359+ζ7314+ζ7313ζ7370+ζ7365+ζ738+ζ733ζ7371+ζ7354+ζ7319+ζ732ζ7361+ζ7341+ζ7332+ζ7312ζ7356+ζ7352+ζ7321+ζ7317ζ7372+ζ7346+ζ7327+ζ73ζ7342+ζ7339+ζ7334+ζ7331ζ7367+ζ7357+ζ7316+ζ736ζ7364+ζ7349+ζ7324+ζ739ζ7368+ζ7362+ζ7311+ζ735ζ7363+ζ7351+ζ7322+ζ7310ζ7358+ζ7340+ζ7333+ζ7315    orthogonal faithful
ρ124000ζ7347+ζ7345+ζ7328+ζ7326ζ7369+ζ7338+ζ7335+ζ734ζ7342+ζ7339+ζ7334+ζ7331ζ7364+ζ7349+ζ7324+ζ739ζ7363+ζ7351+ζ7322+ζ7310ζ7370+ζ7365+ζ738+ζ733ζ7368+ζ7362+ζ7311+ζ735ζ7355+ζ7348+ζ7325+ζ7318ζ7361+ζ7341+ζ7332+ζ7312ζ7372+ζ7346+ζ7327+ζ73ζ7353+ζ7344+ζ7329+ζ7320ζ7367+ζ7357+ζ7316+ζ736ζ7358+ζ7340+ζ7333+ζ7315ζ7350+ζ7337+ζ7336+ζ7323ζ7371+ζ7354+ζ7319+ζ732ζ7366+ζ7343+ζ7330+ζ737ζ7360+ζ7359+ζ7314+ζ7313ζ7356+ζ7352+ζ7321+ζ7317    orthogonal faithful
ρ134000ζ7361+ζ7341+ζ7332+ζ7312ζ7358+ζ7340+ζ7333+ζ7315ζ7355+ζ7348+ζ7325+ζ7318ζ7356+ζ7352+ζ7321+ζ7317ζ7372+ζ7346+ζ7327+ζ73ζ7366+ζ7343+ζ7330+ζ737ζ7350+ζ7337+ζ7336+ζ7323ζ7342+ζ7339+ζ7334+ζ7331ζ7347+ζ7345+ζ7328+ζ7326ζ7363+ζ7351+ζ7322+ζ7310ζ7371+ζ7354+ζ7319+ζ732ζ7360+ζ7359+ζ7314+ζ7313ζ7369+ζ7338+ζ7335+ζ734ζ7368+ζ7362+ζ7311+ζ735ζ7353+ζ7344+ζ7329+ζ7320ζ7370+ζ7365+ζ738+ζ733ζ7367+ζ7357+ζ7316+ζ736ζ7364+ζ7349+ζ7324+ζ739    orthogonal faithful
ρ144000ζ7355+ζ7348+ζ7325+ζ7318ζ7360+ζ7359+ζ7314+ζ7313ζ7372+ζ7346+ζ7327+ζ73ζ7368+ζ7362+ζ7311+ζ735ζ7369+ζ7338+ζ7335+ζ734ζ7347+ζ7345+ζ7328+ζ7326ζ7371+ζ7354+ζ7319+ζ732ζ7363+ζ7351+ζ7322+ζ7310ζ7342+ζ7339+ζ7334+ζ7331ζ7358+ζ7340+ζ7333+ζ7315ζ7370+ζ7365+ζ738+ζ733ζ7356+ζ7352+ζ7321+ζ7317ζ7367+ζ7357+ζ7316+ζ736ζ7353+ζ7344+ζ7329+ζ7320ζ7366+ζ7343+ζ7330+ζ737ζ7361+ζ7341+ζ7332+ζ7312ζ7364+ζ7349+ζ7324+ζ739ζ7350+ζ7337+ζ7336+ζ7323    orthogonal faithful
ρ154000ζ7367+ζ7357+ζ7316+ζ736ζ7353+ζ7344+ζ7329+ζ7320ζ7364+ζ7349+ζ7324+ζ739ζ7347+ζ7345+ζ7328+ζ7326ζ7350+ζ7337+ζ7336+ζ7323ζ7358+ζ7340+ζ7333+ζ7315ζ7355+ζ7348+ζ7325+ζ7318ζ7356+ζ7352+ζ7321+ζ7317ζ7360+ζ7359+ζ7314+ζ7313ζ7368+ζ7362+ζ7311+ζ735ζ7372+ζ7346+ζ7327+ζ73ζ7366+ζ7343+ζ7330+ζ737ζ7371+ζ7354+ζ7319+ζ732ζ7342+ζ7339+ζ7334+ζ7331ζ7363+ζ7351+ζ7322+ζ7310ζ7369+ζ7338+ζ7335+ζ734ζ7370+ζ7365+ζ738+ζ733ζ7361+ζ7341+ζ7332+ζ7312    orthogonal faithful
ρ164000ζ7372+ζ7346+ζ7327+ζ73ζ7356+ζ7352+ζ7321+ζ7317ζ7369+ζ7338+ζ7335+ζ734ζ7353+ζ7344+ζ7329+ζ7320ζ7367+ζ7357+ζ7316+ζ736ζ7342+ζ7339+ζ7334+ζ7331ζ7370+ζ7365+ζ738+ζ733ζ7358+ζ7340+ζ7333+ζ7315ζ7363+ζ7351+ζ7322+ζ7310ζ7360+ζ7359+ζ7314+ζ7313ζ7361+ζ7341+ζ7332+ζ7312ζ7368+ζ7362+ζ7311+ζ735ζ7364+ζ7349+ζ7324+ζ739ζ7366+ζ7343+ζ7330+ζ737ζ7347+ζ7345+ζ7328+ζ7326ζ7355+ζ7348+ζ7325+ζ7318ζ7350+ζ7337+ζ7336+ζ7323ζ7371+ζ7354+ζ7319+ζ732    orthogonal faithful
ρ174000ζ7342+ζ7339+ζ7334+ζ7331ζ7367+ζ7357+ζ7316+ζ736ζ7363+ζ7351+ζ7322+ζ7310ζ7350+ζ7337+ζ7336+ζ7323ζ7358+ζ7340+ζ7333+ζ7315ζ7361+ζ7341+ζ7332+ζ7312ζ7353+ζ7344+ζ7329+ζ7320ζ7372+ζ7346+ζ7327+ζ73ζ7355+ζ7348+ζ7325+ζ7318ζ7369+ζ7338+ζ7335+ζ734ζ7366+ζ7343+ζ7330+ζ737ζ7364+ζ7349+ζ7324+ζ739ζ7360+ζ7359+ζ7314+ζ7313ζ7371+ζ7354+ζ7319+ζ732ζ7370+ζ7365+ζ738+ζ733ζ7347+ζ7345+ζ7328+ζ7326ζ7356+ζ7352+ζ7321+ζ7317ζ7368+ζ7362+ζ7311+ζ735    orthogonal faithful
ρ184000ζ7366+ζ7343+ζ7330+ζ737ζ7372+ζ7346+ζ7327+ζ73ζ7347+ζ7345+ζ7328+ζ7326ζ7367+ζ7357+ζ7316+ζ736ζ7342+ζ7339+ζ7334+ζ7331ζ7371+ζ7354+ζ7319+ζ732ζ7356+ζ7352+ζ7321+ζ7317ζ7361+ζ7341+ζ7332+ζ7312ζ7370+ζ7365+ζ738+ζ733ζ7355+ζ7348+ζ7325+ζ7318ζ7368+ζ7362+ζ7311+ζ735ζ7369+ζ7338+ζ7335+ζ734ζ7363+ζ7351+ζ7322+ζ7310ζ7364+ζ7349+ζ7324+ζ739ζ7350+ζ7337+ζ7336+ζ7323ζ7353+ζ7344+ζ7329+ζ7320ζ7358+ζ7340+ζ7333+ζ7315ζ7360+ζ7359+ζ7314+ζ7313    orthogonal faithful
ρ194000ζ7358+ζ7340+ζ7333+ζ7315ζ7350+ζ7337+ζ7336+ζ7323ζ7360+ζ7359+ζ7314+ζ7313ζ7370+ζ7365+ζ738+ζ733ζ7356+ζ7352+ζ7321+ζ7317ζ7372+ζ7346+ζ7327+ζ73ζ7347+ζ7345+ζ7328+ζ7326ζ7367+ζ7357+ζ7316+ζ736ζ7369+ζ7338+ζ7335+ζ734ζ7364+ζ7349+ζ7324+ζ739ζ7342+ζ7339+ζ7334+ζ7331ζ7371+ζ7354+ζ7319+ζ732ζ7368+ζ7362+ζ7311+ζ735ζ7361+ζ7341+ζ7332+ζ7312ζ7355+ζ7348+ζ7325+ζ7318ζ7363+ζ7351+ζ7322+ζ7310ζ7353+ζ7344+ζ7329+ζ7320ζ7366+ζ7343+ζ7330+ζ737    orthogonal faithful
ρ204000ζ7364+ζ7349+ζ7324+ζ739ζ7366+ζ7343+ζ7330+ζ737ζ7350+ζ7337+ζ7336+ζ7323ζ7342+ζ7339+ζ7334+ζ7331ζ7371+ζ7354+ζ7319+ζ732ζ7360+ζ7359+ζ7314+ζ7313ζ7372+ζ7346+ζ7327+ζ73ζ7368+ζ7362+ζ7311+ζ735ζ7356+ζ7352+ζ7321+ζ7317ζ7353+ζ7344+ζ7329+ζ7320ζ7369+ζ7338+ζ7335+ζ734ζ7347+ζ7345+ζ7328+ζ7326ζ7370+ζ7365+ζ738+ζ733ζ7363+ζ7351+ζ7322+ζ7310ζ7358+ζ7340+ζ7333+ζ7315ζ7367+ζ7357+ζ7316+ζ736ζ7361+ζ7341+ζ7332+ζ7312ζ7355+ζ7348+ζ7325+ζ7318    orthogonal faithful
ρ214000ζ7368+ζ7362+ζ7311+ζ735ζ7361+ζ7341+ζ7332+ζ7312ζ7353+ζ7344+ζ7329+ζ7320ζ7372+ζ7346+ζ7327+ζ73ζ7366+ζ7343+ζ7330+ζ737ζ7364+ζ7349+ζ7324+ζ739ζ7358+ζ7340+ζ7333+ζ7315ζ7371+ζ7354+ζ7319+ζ732ζ7350+ζ7337+ζ7336+ζ7323ζ7370+ζ7365+ζ738+ζ733ζ7360+ζ7359+ζ7314+ζ7313ζ7355+ζ7348+ζ7325+ζ7318ζ7347+ζ7345+ζ7328+ζ7326ζ7369+ζ7338+ζ7335+ζ734ζ7367+ζ7357+ζ7316+ζ736ζ7356+ζ7352+ζ7321+ζ7317ζ7342+ζ7339+ζ7334+ζ7331ζ7363+ζ7351+ζ7322+ζ7310    orthogonal faithful
ρ224000ζ7370+ζ7365+ζ738+ζ733ζ7363+ζ7351+ζ7322+ζ7310ζ7361+ζ7341+ζ7332+ζ7312ζ7360+ζ7359+ζ7314+ζ7313ζ7355+ζ7348+ζ7325+ζ7318ζ7353+ζ7344+ζ7329+ζ7320ζ7364+ζ7349+ζ7324+ζ739ζ7347+ζ7345+ζ7328+ζ7326ζ7366+ζ7343+ζ7330+ζ737ζ7342+ζ7339+ζ7334+ζ7331ζ7350+ζ7337+ζ7336+ζ7323ζ7358+ζ7340+ζ7333+ζ7315ζ7372+ζ7346+ζ7327+ζ73ζ7356+ζ7352+ζ7321+ζ7317ζ7368+ζ7362+ζ7311+ζ735ζ7371+ζ7354+ζ7319+ζ732ζ7369+ζ7338+ζ7335+ζ734ζ7367+ζ7357+ζ7316+ζ736    orthogonal faithful

Smallest permutation representation of C73⋊C4
►On 73 points: primitive
Generators in S73
(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)
(2 28 73 47)(3 55 72 20)(4 9 71 66)(5 36 70 39)(6 63 69 12)(7 17 68 58)(8 44 67 31)(10 25 65 50)(11 52 64 23)(13 33 62 42)(14 60 61 15)(16 41 59 34)(18 22 57 53)(19 49 56 26)(21 30 54 45)(24 38 51 37)(27 46 48 29)(32 35 43 40)
 
G:=sub<Sym(73)| (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), (2,28,73,47)(3,55,72,20)(4,9,71,66)(5,36,70,39)(6,63,69,12)(7,17,68,58)(8,44,67,31)(10,25,65,50)(11,52,64,23)(13,33,62,42)(14,60,61,15)(16,41,59,34)(18,22,57,53)(19,49,56,26)(21,30,54,45)(24,38,51,37)(27,46,48,29)(32,35,43,40)>;
 
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), (2,28,73,47)(3,55,72,20)(4,9,71,66)(5,36,70,39)(6,63,69,12)(7,17,68,58)(8,44,67,31)(10,25,65,50)(11,52,64,23)(13,33,62,42)(14,60,61,15)(16,41,59,34)(18,22,57,53)(19,49,56,26)(21,30,54,45)(24,38,51,37)(27,46,48,29)(32,35,43,40) );
 
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)], [(2,28,73,47),(3,55,72,20),(4,9,71,66),(5,36,70,39),(6,63,69,12),(7,17,68,58),(8,44,67,31),(10,25,65,50),(11,52,64,23),(13,33,62,42),(14,60,61,15),(16,41,59,34),(18,22,57,53),(19,49,56,26),(21,30,54,45),(24,38,51,37),(27,46,48,29),(32,35,43,40)]])
 

Matrix representation of C73⋊C4 ►in GL4(𝔽293) generated by

74100
259010
120001
15458196170
,
107323187
551719943
191262101277
568280207
G:=sub<GL(4,GF(293))| [74,259,120,154,1,0,0,58,0,1,0,196,0,0,1,170],[107,55,191,56,32,171,262,82,31,99,101,80,87,43,277,207] >;
 

C73⋊C4 in GAP, Magma, Sage, TeX

C_{73}\rtimes C_4
 
% in TeX
 
G:=Group("C73:C4");
 
// GroupNames label
 
G:=SmallGroup(292,3);
 
// by ID
 
G=gap.SmallGroup(292,3);
 
# by ID
 
G:=PCGroup([3,-2,-2,-73,6,974,1301]);
 
// Polycyclic
 
G:=Group<a,b|a^73=b^4=1,b*a*b^-1=a^46>;
 
// generators/relations
 

Export

Subgroup lattice of C73⋊C4 in TeX
Character table of C73⋊C4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁