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G = C73⋊C3order 219 = 3·73

The semidirect product of C73 and C3 acting faithfully

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C73⋊C3, SmallGroup(219,1)

Series: Derived Chief Lower central Upper central

C1C73 — C73⋊C3
C1C73 — C73⋊C3
C73 — C73⋊C3
C1

Generators and relations for C73⋊C3
 G = < a,b | a73=b3=1, bab-1=a64 >

73C3

Character table of C73⋊C3

 class 13A3B73A73B73C73D73E73F73G73H73I73J73K73L73M73N73O73P73Q73R73S73T73U73V73W73X
 size 17373333333333333333333333333
ρ1111111111111111111111111111    trivial
ρ21ζ32ζ3111111111111111111111111    linear of order 3
ρ31ζ3ζ32111111111111111111111111    linear of order 3
ρ4300ζ73377332734ζ736773547325ζ733173297313ζ736473873ζ736273587326ζ733873237312ζ733973207314ζ73557316732ζ736673637317ζ735273517343ζ734773157311ζ73467324733ζ73407328735ζ73727365739ζ735973537334ζ736973417336ζ736073447342ζ733073227321ζ737173577318ζ73487319736ζ73567310737ζ736873457333ζ736173507335ζ737073497327    complex faithful
ρ5300ζ733873237312ζ73557316732ζ733973207314ζ73467324733ζ73407328735ζ736973417336ζ736073447342ζ73487319736ζ735273517343ζ73567310737ζ736873457333ζ73727365739ζ734773157311ζ737073497327ζ733173297313ζ736173507335ζ735973537334ζ736673637317ζ736773547325ζ737173577318ζ733073227321ζ736273587326ζ73377332734ζ736473873    complex faithful
ρ6300ζ736173507335ζ737173577318ζ735973537334ζ737073497327ζ736873457333ζ73377332734ζ733173297313ζ736773547325ζ733073227321ζ736673637317ζ73407328735ζ736473873ζ736273587326ζ73467324733ζ736073447342ζ733873237312ζ733973207314ζ73567310737ζ73487319736ζ73557316732ζ735273517343ζ734773157311ζ736973417336ζ73727365739    complex faithful
ρ7300ζ735973537334ζ733073227321ζ736473873ζ736873457333ζ73557316732ζ733173297313ζ73467324733ζ736673637317ζ736173507335ζ73377332734ζ737173577318ζ736273587326ζ73487319736ζ73407328735ζ737073497327ζ733973207314ζ73727365739ζ736973417336ζ73567310737ζ735273517343ζ733873237312ζ736773547325ζ736073447342ζ734773157311    complex faithful
ρ8300ζ734773157311ζ733973207314ζ736773547325ζ733073227321ζ736173507335ζ736873457333ζ73557316732ζ736073447342ζ73727365739ζ737073497327ζ733873237312ζ736673637317ζ73377332734ζ735273517343ζ737173577318ζ736273587326ζ73487319736ζ73467324733ζ733173297313ζ735973537334ζ736473873ζ736973417336ζ73407328735ζ73567310737    complex faithful
ρ9300ζ73407328735ζ733173297313ζ737173577318ζ73567310737ζ736973417336ζ734773157311ζ736773547325ζ733973207314ζ73467324733ζ73727365739ζ73377332734ζ733073227321ζ736173507335ζ736673637317ζ73487319736ζ736873457333ζ73557316732ζ736473873ζ735973537334ζ736073447342ζ737073497327ζ733873237312ζ736273587326ζ735273517343    complex faithful
ρ10300ζ737173577318ζ73467324733ζ733073227321ζ736973417336ζ736073447342ζ736773547325ζ736673637317ζ73727365739ζ73407328735ζ734773157311ζ733173297313ζ736173507335ζ735973537334ζ73377332734ζ73567310737ζ73557316732ζ735273517343ζ736273587326ζ736473873ζ737073497327ζ736873457333ζ733973207314ζ73487319736ζ733873237312    complex faithful
ρ11300ζ736773547325ζ73727365739ζ736673637317ζ736173507335ζ735973537334ζ73557316732ζ735273517343ζ737073497327ζ734773157311ζ736873457333ζ733973207314ζ73377332734ζ733173297313ζ733873237312ζ733073227321ζ73487319736ζ73567310737ζ73407328735ζ73467324733ζ736473873ζ736273587326ζ736073447342ζ737173577318ζ736973417336    complex faithful
ρ12300ζ733073227321ζ73407328735ζ736173507335ζ736073447342ζ737073497327ζ736673637317ζ73377332734ζ734773157311ζ737173577318ζ736773547325ζ73467324733ζ735973537334ζ736473873ζ733173297313ζ736973417336ζ735273517343ζ733873237312ζ73487319736ζ736273587326ζ736873457333ζ73557316732ζ73727365739ζ73567310737ζ733973207314    complex faithful
ρ13300ζ736273587326ζ735973537334ζ73487319736ζ735273517343ζ733873237312ζ73407328735ζ737173577318ζ733173297313ζ736473873ζ73467324733ζ736173507335ζ73567310737ζ736973417336ζ733073227321ζ73557316732ζ734773157311ζ736773547325ζ737073497327ζ736073447342ζ733973207314ζ73727365739ζ73377332734ζ736873457333ζ736673637317    complex faithful
ρ14300ζ736873457333ζ736073447342ζ73557316732ζ736673637317ζ73377332734ζ736273587326ζ73487319736ζ735973537334ζ737073497327ζ736473873ζ736973417336ζ735273517343ζ733873237312ζ73567310737ζ736773547325ζ73407328735ζ737173577318ζ73727365739ζ733973207314ζ733173297313ζ73467324733ζ736173507335ζ734773157311ζ733073227321    complex faithful
ρ15300ζ736973417336ζ73487319736ζ736073447342ζ73727365739ζ734773157311ζ736173507335ζ735973537334ζ737173577318ζ73567310737ζ733073227321ζ736273587326ζ737073497327ζ736873457333ζ736473873ζ733973207314ζ73377332734ζ733173297313ζ735273517343ζ73557316732ζ736773547325ζ736673637317ζ73407328735ζ733873237312ζ73467324733    complex faithful
ρ16300ζ73567310737ζ736273587326ζ736973417336ζ733973207314ζ73727365739ζ733073227321ζ736173507335ζ73407328735ζ73487319736ζ737173577318ζ736473873ζ736073447342ζ737073497327ζ735973537334ζ733873237312ζ736673637317ζ73377332734ζ73557316732ζ736873457333ζ734773157311ζ736773547325ζ73467324733ζ735273517343ζ733173297313    complex faithful
ρ17300ζ73467324733ζ73377332734ζ73407328735ζ73487319736ζ73567310737ζ73727365739ζ734773157311ζ733873237312ζ733173297313ζ733973207314ζ736673637317ζ737173577318ζ733073227321ζ736773547325ζ736273587326ζ737073497327ζ736873457333ζ735973537334ζ736173507335ζ736973417336ζ736073447342ζ735273517343ζ736473873ζ73557316732    complex faithful
ρ18300ζ736673637317ζ734773157311ζ73377332734ζ735973537334ζ736473873ζ735273517343ζ733873237312ζ736873457333ζ736773547325ζ73557316732ζ73727365739ζ733173297313ζ73467324733ζ733973207314ζ736173507335ζ73567310737ζ736973417336ζ737173577318ζ73407328735ζ736273587326ζ73487319736ζ737073497327ζ733073227321ζ736073447342    complex faithful
ρ19300ζ736473873ζ736173507335ζ736273587326ζ73557316732ζ735273517343ζ73467324733ζ73407328735ζ73377332734ζ735973537334ζ733173297313ζ733073227321ζ73487319736ζ73567310737ζ737173577318ζ736873457333ζ73727365739ζ734773157311ζ736073447342ζ736973417336ζ733873237312ζ733973207314ζ736673637317ζ737073497327ζ736773547325    complex faithful
ρ20300ζ735273517343ζ736873457333ζ733873237312ζ733173297313ζ73467324733ζ73567310737ζ736973417336ζ736273587326ζ73557316732ζ73487319736ζ737073497327ζ733973207314ζ73727365739ζ736073447342ζ73377332734ζ733073227321ζ736173507335ζ736773547325ζ734773157311ζ73407328735ζ737173577318ζ736473873ζ736673637317ζ735973537334    complex faithful
ρ21300ζ733973207314ζ735273517343ζ73727365739ζ73407328735ζ737173577318ζ736073447342ζ737073497327ζ73567310737ζ733873237312ζ736973417336ζ73557316732ζ734773157311ζ736773547325ζ736873457333ζ73467324733ζ735973537334ζ736473873ζ73377332734ζ736673637317ζ733073227321ζ736173507335ζ73487319736ζ733173297313ζ736273587326    complex faithful
ρ22300ζ73487319736ζ736473873ζ73567310737ζ733873237312ζ733973207314ζ737173577318ζ733073227321ζ73467324733ζ736273587326ζ73407328735ζ735973537334ζ736973417336ζ736073447342ζ736173507335ζ735273517343ζ736773547325ζ736673637317ζ736873457333ζ737073497327ζ73727365739ζ734773157311ζ733173297313ζ73557316732ζ73377332734    complex faithful
ρ23300ζ73557316732ζ737073497327ζ735273517343ζ73377332734ζ733173297313ζ73487319736ζ73567310737ζ736473873ζ736873457333ζ736273587326ζ736073447342ζ733873237312ζ733973207314ζ736973417336ζ736673637317ζ737173577318ζ733073227321ζ734773157311ζ73727365739ζ73467324733ζ73407328735ζ735973537334ζ736773547325ζ736173507335    complex faithful
ρ24300ζ736073447342ζ73567310737ζ737073497327ζ734773157311ζ736773547325ζ735973537334ζ736473873ζ733073227321ζ736973417336ζ736173507335ζ73487319736ζ736873457333ζ73557316732ζ736273587326ζ73727365739ζ733173297313ζ73467324733ζ733873237312ζ735273517343ζ736673637317ζ73377332734ζ737173577318ζ733973207314ζ73407328735    complex faithful
ρ25300ζ733173297313ζ736673637317ζ73467324733ζ736273587326ζ73487319736ζ733973207314ζ73727365739ζ735273517343ζ73377332734ζ733873237312ζ736773547325ζ73407328735ζ737173577318ζ734773157311ζ736473873ζ736073447342ζ737073497327ζ736173507335ζ733073227321ζ73567310737ζ736973417336ζ73557316732ζ735973537334ζ736873457333    complex faithful
ρ26300ζ73727365739ζ733873237312ζ734773157311ζ737173577318ζ733073227321ζ737073497327ζ736873457333ζ736973417336ζ733973207314ζ736073447342ζ735273517343ζ736773547325ζ736673637317ζ73557316732ζ73407328735ζ736473873ζ736273587326ζ733173297313ζ73377332734ζ736173507335ζ735973537334ζ73567310737ζ73467324733ζ73487319736    complex faithful
ρ27300ζ737073497327ζ736973417336ζ736873457333ζ736773547325ζ736673637317ζ736473873ζ736273587326ζ736173507335ζ736073447342ζ735973537334ζ73567310737ζ73557316732ζ735273517343ζ73487319736ζ734773157311ζ73467324733ζ73407328735ζ733973207314ζ733873237312ζ73377332734ζ733173297313ζ733073227321ζ73727365739ζ737173577318    complex faithful

Smallest permutation representation of C73⋊C3
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 9 65)(3 17 56)(4 25 47)(5 33 38)(6 41 29)(7 49 20)(8 57 11)(10 73 66)(12 16 48)(13 24 39)(14 32 30)(15 40 21)(18 64 67)(19 72 58)(22 23 31)(26 55 68)(27 63 59)(28 71 50)(34 46 69)(35 54 60)(36 62 51)(37 70 42)(43 45 61)(44 53 52)

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

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

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

C73⋊C3 is a maximal subgroup of   C73⋊C6

Matrix representation of C73⋊C3 in GL3(𝔽439) generated by

13010
43301
212220122
,
27104113
386303118
416242109
G:=sub<GL(3,GF(439))| [130,433,212,1,0,220,0,1,122],[27,386,416,104,303,242,113,118,109] >;

C73⋊C3 in GAP, Magma, Sage, TeX

C_{73}\rtimes C_3
% in TeX

G:=Group("C73:C3");
// GroupNames label

G:=SmallGroup(219,1);
// by ID

G=gap.SmallGroup(219,1);
# by ID

G:=PCGroup([2,-3,-73,97]);
// Polycyclic

G:=Group<a,b|a^73=b^3=1,b*a*b^-1=a^64>;
// generators/relations

Export

Subgroup lattice of C73⋊C3 in TeX
Character table of C73⋊C3 in TeX

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