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G = C73⋊C4order 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

C1C73 — C73⋊C4
C1C73D73 — 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
ρ54000ζ736973387335734ζ736873627311735ζ736773577316736ζ736673437330737ζ736473497324739ζ7363735173227310ζ7361734173327312ζ7360735973147313ζ7358734073337315ζ7356735273217317ζ7355734873257318ζ7353734473297320ζ7350733773367323ζ7347734573287326ζ7342733973347331ζ73727346732773ζ737173547319732ζ73707365738733    orthogonal faithful
ρ64000ζ7350733773367323ζ7347734573287326ζ737173547319732ζ7363735173227310ζ73707365738733ζ7356735273217317ζ736973387335734ζ7353734473297320ζ736873627311735ζ736673437330737ζ736773577316736ζ7342733973347331ζ7361734173327312ζ7358734073337315ζ7360735973147313ζ736473497324739ζ7355734873257318ζ73727346732773    orthogonal faithful
ρ74000ζ7356735273217317ζ73707365738733ζ736873627311735ζ7355734873257318ζ7353734473297320ζ736773577316736ζ7363735173227310ζ7350733773367323ζ736473497324739ζ737173547319732ζ7358734073337315ζ7361734173327312ζ736673437330737ζ73727346732773ζ736973387335734ζ7360735973147313ζ7347734573287326ζ7342733973347331    orthogonal faithful
ρ84000ζ737173547319732ζ7342733973347331ζ73707365738733ζ7358734073337315ζ7361734173327312ζ736873627311735ζ736773577316736ζ736673437330737ζ7353734473297320ζ7347734573287326ζ736473497324739ζ7363735173227310ζ7355734873257318ζ7360735973147313ζ7356735273217317ζ7350733773367323ζ73727346732773ζ736973387335734    orthogonal faithful
ρ94000ζ7360735973147313ζ737173547319732ζ7356735273217317ζ7361734173327312ζ736873627311735ζ736973387335734ζ7342733973347331ζ736473497324739ζ736773577316736ζ7350733773367323ζ7363735173227310ζ73707365738733ζ7353734473297320ζ7355734873257318ζ73727346732773ζ7358734073337315ζ736673437330737ζ7347734573287326    orthogonal faithful
ρ104000ζ7363735173227310ζ736473497324739ζ7358734073337315ζ737173547319732ζ7360735973147313ζ7355734873257318ζ736673437330737ζ736973387335734ζ73727346732773ζ736773577316736ζ7347734573287326ζ7350733773367323ζ7356735273217317ζ73707365738733ζ7361734173327312ζ7342733973347331ζ736873627311735ζ7353734473297320    orthogonal faithful
ρ114000ζ7353734473297320ζ7355734873257318ζ736673437330737ζ736973387335734ζ7347734573287326ζ7350733773367323ζ7360735973147313ζ73707365738733ζ737173547319732ζ7361734173327312ζ7356735273217317ζ73727346732773ζ7342733973347331ζ736773577316736ζ736473497324739ζ736873627311735ζ7363735173227310ζ7358734073337315    orthogonal faithful
ρ124000ζ7347734573287326ζ736973387335734ζ7342733973347331ζ736473497324739ζ7363735173227310ζ73707365738733ζ736873627311735ζ7355734873257318ζ7361734173327312ζ73727346732773ζ7353734473297320ζ736773577316736ζ7358734073337315ζ7350733773367323ζ737173547319732ζ736673437330737ζ7360735973147313ζ7356735273217317    orthogonal faithful
ρ134000ζ7361734173327312ζ7358734073337315ζ7355734873257318ζ7356735273217317ζ73727346732773ζ736673437330737ζ7350733773367323ζ7342733973347331ζ7347734573287326ζ7363735173227310ζ737173547319732ζ7360735973147313ζ736973387335734ζ736873627311735ζ7353734473297320ζ73707365738733ζ736773577316736ζ736473497324739    orthogonal faithful
ρ144000ζ7355734873257318ζ7360735973147313ζ73727346732773ζ736873627311735ζ736973387335734ζ7347734573287326ζ737173547319732ζ7363735173227310ζ7342733973347331ζ7358734073337315ζ73707365738733ζ7356735273217317ζ736773577316736ζ7353734473297320ζ736673437330737ζ7361734173327312ζ736473497324739ζ7350733773367323    orthogonal faithful
ρ154000ζ736773577316736ζ7353734473297320ζ736473497324739ζ7347734573287326ζ7350733773367323ζ7358734073337315ζ7355734873257318ζ7356735273217317ζ7360735973147313ζ736873627311735ζ73727346732773ζ736673437330737ζ737173547319732ζ7342733973347331ζ7363735173227310ζ736973387335734ζ73707365738733ζ7361734173327312    orthogonal faithful
ρ164000ζ73727346732773ζ7356735273217317ζ736973387335734ζ7353734473297320ζ736773577316736ζ7342733973347331ζ73707365738733ζ7358734073337315ζ7363735173227310ζ7360735973147313ζ7361734173327312ζ736873627311735ζ736473497324739ζ736673437330737ζ7347734573287326ζ7355734873257318ζ7350733773367323ζ737173547319732    orthogonal faithful
ρ174000ζ7342733973347331ζ736773577316736ζ7363735173227310ζ7350733773367323ζ7358734073337315ζ7361734173327312ζ7353734473297320ζ73727346732773ζ7355734873257318ζ736973387335734ζ736673437330737ζ736473497324739ζ7360735973147313ζ737173547319732ζ73707365738733ζ7347734573287326ζ7356735273217317ζ736873627311735    orthogonal faithful
ρ184000ζ736673437330737ζ73727346732773ζ7347734573287326ζ736773577316736ζ7342733973347331ζ737173547319732ζ7356735273217317ζ7361734173327312ζ73707365738733ζ7355734873257318ζ736873627311735ζ736973387335734ζ7363735173227310ζ736473497324739ζ7350733773367323ζ7353734473297320ζ7358734073337315ζ7360735973147313    orthogonal faithful
ρ194000ζ7358734073337315ζ7350733773367323ζ7360735973147313ζ73707365738733ζ7356735273217317ζ73727346732773ζ7347734573287326ζ736773577316736ζ736973387335734ζ736473497324739ζ7342733973347331ζ737173547319732ζ736873627311735ζ7361734173327312ζ7355734873257318ζ7363735173227310ζ7353734473297320ζ736673437330737    orthogonal faithful
ρ204000ζ736473497324739ζ736673437330737ζ7350733773367323ζ7342733973347331ζ737173547319732ζ7360735973147313ζ73727346732773ζ736873627311735ζ7356735273217317ζ7353734473297320ζ736973387335734ζ7347734573287326ζ73707365738733ζ7363735173227310ζ7358734073337315ζ736773577316736ζ7361734173327312ζ7355734873257318    orthogonal faithful
ρ214000ζ736873627311735ζ7361734173327312ζ7353734473297320ζ73727346732773ζ736673437330737ζ736473497324739ζ7358734073337315ζ737173547319732ζ7350733773367323ζ73707365738733ζ7360735973147313ζ7355734873257318ζ7347734573287326ζ736973387335734ζ736773577316736ζ7356735273217317ζ7342733973347331ζ7363735173227310    orthogonal faithful
ρ224000ζ73707365738733ζ7363735173227310ζ7361734173327312ζ7360735973147313ζ7355734873257318ζ7353734473297320ζ736473497324739ζ7347734573287326ζ736673437330737ζ7342733973347331ζ7350733773367323ζ7358734073337315ζ73727346732773ζ7356735273217317ζ736873627311735ζ737173547319732ζ736973387335734ζ736773577316736    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

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