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G = C192C9order 171 = 32·19

The semidirect product of C19 and C9 acting via C9/C3=C3

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

Aliases: C192C9, C57.C3, C3.(C19⋊C3), SmallGroup(171,1)

Series: Derived Chief Lower central Upper central

C1C19 — C192C9
C1C19C57 — C192C9
C19 — C192C9
C1C3

Generators and relations for C192C9
 G = < a,b | a19=b9=1, bab-1=a11 >

19C9

Character table of C192C9

 class 13A3B9A9B9C9D9E9F19A19B19C19D19E19F57A57B57C57D57E57F57G57H57I57J57K57L
 size 111191919191919333333333333333333
ρ1111111111111111111111111111    trivial
ρ2111ζ32ζ3ζ3ζ32ζ32ζ3111111111111111111    linear of order 3
ρ3111ζ3ζ32ζ32ζ3ζ3ζ32111111111111111111    linear of order 3
ρ41ζ32ζ3ζ9ζ95ζ92ζ97ζ94ζ98111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ32    linear of order 9
ρ51ζ3ζ32ζ98ζ94ζ97ζ92ζ95ζ9111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ3    linear of order 9
ρ61ζ32ζ3ζ94ζ92ζ98ζ9ζ97ζ95111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ32    linear of order 9
ρ71ζ3ζ32ζ95ζ97ζ9ζ98ζ92ζ94111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ3    linear of order 9
ρ81ζ3ζ32ζ92ζ9ζ94ζ95ζ98ζ97111111ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ3    linear of order 9
ρ91ζ32ζ3ζ97ζ98ζ95ζ94ζ9ζ92111111ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ32    linear of order 9
ρ10333000000ζ191119719ζ1914193192ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ1914193192ζ1914193192ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ19171916195ζ19181912198ζ191519131910ζ199196194ζ191119719ζ191119719    complex lifted from C19⋊C3
ρ11333000000ζ199196194ζ19181912198ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ19181912198ζ19181912198ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ191119719ζ191519131910ζ1914193192ζ19171916195ζ199196194ζ199196194    complex lifted from C19⋊C3
ρ12333000000ζ191519131910ζ191119719ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ191119719ζ191119719ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ19181912198ζ199196194ζ19171916195ζ1914193192ζ191519131910ζ191519131910    complex lifted from C19⋊C3
ρ13333000000ζ19171916195ζ191519131910ζ191119719ζ199196194ζ1914193192ζ19181912198ζ191519131910ζ191519131910ζ191119719ζ199196194ζ1914193192ζ19181912198ζ199196194ζ1914193192ζ19181912198ζ191119719ζ19171916195ζ19171916195    complex lifted from C19⋊C3
ρ14333000000ζ1914193192ζ199196194ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ199196194ζ199196194ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ191519131910ζ19171916195ζ191119719ζ19181912198ζ1914193192ζ1914193192    complex lifted from C19⋊C3
ρ15333000000ζ19181912198ζ19171916195ζ191519131910ζ1914193192ζ191119719ζ199196194ζ19171916195ζ19171916195ζ191519131910ζ1914193192ζ191119719ζ199196194ζ1914193192ζ191119719ζ199196194ζ191519131910ζ19181912198ζ19181912198    complex lifted from C19⋊C3
ρ163-3+3-3/2-3-3-3/2000000ζ191519131910ζ191119719ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ3ζ19113ζ1973ζ19ζ32ζ191132ζ19732ζ19ζ32ζ191432ζ19332ζ192ζ32ζ191832ζ191232ζ198ζ32ζ19932ζ19632ζ194ζ32ζ191732ζ191632ζ195ζ3ζ19183ζ19123ζ198ζ3ζ1993ζ1963ζ194ζ3ζ19173ζ19163ζ195ζ3ζ19143ζ1933ζ192ζ32ζ191532ζ191332ζ1910ζ3ζ19153ζ19133ζ1910    complex faithful
ρ173-3-3-3/2-3+3-3/2000000ζ1914193192ζ199196194ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ32ζ19932ζ19632ζ194ζ3ζ1993ζ1963ζ194ζ3ζ19183ζ19123ζ198ζ3ζ19153ζ19133ζ1910ζ3ζ19173ζ19163ζ195ζ3ζ19113ζ1973ζ19ζ32ζ191532ζ191332ζ1910ζ32ζ191732ζ191632ζ195ζ32ζ191132ζ19732ζ19ζ32ζ191832ζ191232ζ198ζ3ζ19143ζ1933ζ192ζ32ζ191432ζ19332ζ192    complex faithful
ρ183-3+3-3/2-3-3-3/2000000ζ19171916195ζ191519131910ζ191119719ζ199196194ζ1914193192ζ19181912198ζ3ζ19153ζ19133ζ1910ζ32ζ191532ζ191332ζ1910ζ32ζ191132ζ19732ζ19ζ32ζ19932ζ19632ζ194ζ32ζ191432ζ19332ζ192ζ32ζ191832ζ191232ζ198ζ3ζ1993ζ1963ζ194ζ3ζ19143ζ1933ζ192ζ3ζ19183ζ19123ζ198ζ3ζ19113ζ1973ζ19ζ32ζ191732ζ191632ζ195ζ3ζ19173ζ19163ζ195    complex faithful
ρ193-3+3-3/2-3-3-3/2000000ζ19181912198ζ19171916195ζ191519131910ζ1914193192ζ191119719ζ199196194ζ3ζ19173ζ19163ζ195ζ32ζ191732ζ191632ζ195ζ32ζ191532ζ191332ζ1910ζ32ζ191432ζ19332ζ192ζ32ζ191132ζ19732ζ19ζ32ζ19932ζ19632ζ194ζ3ζ19143ζ1933ζ192ζ3ζ19113ζ1973ζ19ζ3ζ1993ζ1963ζ194ζ3ζ19153ζ19133ζ1910ζ32ζ191832ζ191232ζ198ζ3ζ19183ζ19123ζ198    complex faithful
ρ203-3+3-3/2-3-3-3/2000000ζ191119719ζ1914193192ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ3ζ19143ζ1933ζ192ζ32ζ191432ζ19332ζ192ζ32ζ19932ζ19632ζ194ζ32ζ191732ζ191632ζ195ζ32ζ191832ζ191232ζ198ζ32ζ191532ζ191332ζ1910ζ3ζ19173ζ19163ζ195ζ3ζ19183ζ19123ζ198ζ3ζ19153ζ19133ζ1910ζ3ζ1993ζ1963ζ194ζ32ζ191132ζ19732ζ19ζ3ζ19113ζ1973ζ19    complex faithful
ρ213-3-3-3/2-3+3-3/2000000ζ199196194ζ19181912198ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ32ζ191832ζ191232ζ198ζ3ζ19183ζ19123ζ198ζ3ζ19173ζ19163ζ195ζ3ζ19113ζ1973ζ19ζ3ζ19153ζ19133ζ1910ζ3ζ19143ζ1933ζ192ζ32ζ191132ζ19732ζ19ζ32ζ191532ζ191332ζ1910ζ32ζ191432ζ19332ζ192ζ32ζ191732ζ191632ζ195ζ3ζ1993ζ1963ζ194ζ32ζ19932ζ19632ζ194    complex faithful
ρ223-3-3-3/2-3+3-3/2000000ζ19171916195ζ191519131910ζ191119719ζ199196194ζ1914193192ζ19181912198ζ32ζ191532ζ191332ζ1910ζ3ζ19153ζ19133ζ1910ζ3ζ19113ζ1973ζ19ζ3ζ1993ζ1963ζ194ζ3ζ19143ζ1933ζ192ζ3ζ19183ζ19123ζ198ζ32ζ19932ζ19632ζ194ζ32ζ191432ζ19332ζ192ζ32ζ191832ζ191232ζ198ζ32ζ191132ζ19732ζ19ζ3ζ19173ζ19163ζ195ζ32ζ191732ζ191632ζ195    complex faithful
ρ233-3-3-3/2-3+3-3/2000000ζ191119719ζ1914193192ζ199196194ζ19171916195ζ19181912198ζ191519131910ζ32ζ191432ζ19332ζ192ζ3ζ19143ζ1933ζ192ζ3ζ1993ζ1963ζ194ζ3ζ19173ζ19163ζ195ζ3ζ19183ζ19123ζ198ζ3ζ19153ζ19133ζ1910ζ32ζ191732ζ191632ζ195ζ32ζ191832ζ191232ζ198ζ32ζ191532ζ191332ζ1910ζ32ζ19932ζ19632ζ194ζ3ζ19113ζ1973ζ19ζ32ζ191132ζ19732ζ19    complex faithful
ρ243-3-3-3/2-3+3-3/2000000ζ19181912198ζ19171916195ζ191519131910ζ1914193192ζ191119719ζ199196194ζ32ζ191732ζ191632ζ195ζ3ζ19173ζ19163ζ195ζ3ζ19153ζ19133ζ1910ζ3ζ19143ζ1933ζ192ζ3ζ19113ζ1973ζ19ζ3ζ1993ζ1963ζ194ζ32ζ191432ζ19332ζ192ζ32ζ191132ζ19732ζ19ζ32ζ19932ζ19632ζ194ζ32ζ191532ζ191332ζ1910ζ3ζ19183ζ19123ζ198ζ32ζ191832ζ191232ζ198    complex faithful
ρ253-3+3-3/2-3-3-3/2000000ζ199196194ζ19181912198ζ19171916195ζ191119719ζ191519131910ζ1914193192ζ3ζ19183ζ19123ζ198ζ32ζ191832ζ191232ζ198ζ32ζ191732ζ191632ζ195ζ32ζ191132ζ19732ζ19ζ32ζ191532ζ191332ζ1910ζ32ζ191432ζ19332ζ192ζ3ζ19113ζ1973ζ19ζ3ζ19153ζ19133ζ1910ζ3ζ19143ζ1933ζ192ζ3ζ19173ζ19163ζ195ζ32ζ19932ζ19632ζ194ζ3ζ1993ζ1963ζ194    complex faithful
ρ263-3+3-3/2-3-3-3/2000000ζ1914193192ζ199196194ζ19181912198ζ191519131910ζ19171916195ζ191119719ζ3ζ1993ζ1963ζ194ζ32ζ19932ζ19632ζ194ζ32ζ191832ζ191232ζ198ζ32ζ191532ζ191332ζ1910ζ32ζ191732ζ191632ζ195ζ32ζ191132ζ19732ζ19ζ3ζ19153ζ19133ζ1910ζ3ζ19173ζ19163ζ195ζ3ζ19113ζ1973ζ19ζ3ζ19183ζ19123ζ198ζ32ζ191432ζ19332ζ192ζ3ζ19143ζ1933ζ192    complex faithful
ρ273-3-3-3/2-3+3-3/2000000ζ191519131910ζ191119719ζ1914193192ζ19181912198ζ199196194ζ19171916195ζ32ζ191132ζ19732ζ19ζ3ζ19113ζ1973ζ19ζ3ζ19143ζ1933ζ192ζ3ζ19183ζ19123ζ198ζ3ζ1993ζ1963ζ194ζ3ζ19173ζ19163ζ195ζ32ζ191832ζ191232ζ198ζ32ζ19932ζ19632ζ194ζ32ζ191732ζ191632ζ195ζ32ζ191432ζ19332ζ192ζ3ζ19153ζ19133ζ1910ζ32ζ191532ζ191332ζ1910    complex faithful

Smallest permutation representation of C192C9
Regular action on 171 points
Generators in S171
(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 90 91 92 93 94 95)(96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114)(115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133)(134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171)
(1 161 107 39 144 81 29 127 76)(2 168 99 40 151 92 30 115 68)(3 156 110 41 139 84 31 122 60)(4 163 102 42 146 95 32 129 71)(5 170 113 43 134 87 33 117 63)(6 158 105 44 141 79 34 124 74)(7 165 97 45 148 90 35 131 66)(8 153 108 46 136 82 36 119 58)(9 160 100 47 143 93 37 126 69)(10 167 111 48 150 85 38 133 61)(11 155 103 49 138 77 20 121 72)(12 162 114 50 145 88 21 128 64)(13 169 106 51 152 80 22 116 75)(14 157 98 52 140 91 23 123 67)(15 164 109 53 147 83 24 130 59)(16 171 101 54 135 94 25 118 70)(17 159 112 55 142 86 26 125 62)(18 166 104 56 149 78 27 132 73)(19 154 96 57 137 89 28 120 65)

G:=sub<Sym(171)| (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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171), (1,161,107,39,144,81,29,127,76)(2,168,99,40,151,92,30,115,68)(3,156,110,41,139,84,31,122,60)(4,163,102,42,146,95,32,129,71)(5,170,113,43,134,87,33,117,63)(6,158,105,44,141,79,34,124,74)(7,165,97,45,148,90,35,131,66)(8,153,108,46,136,82,36,119,58)(9,160,100,47,143,93,37,126,69)(10,167,111,48,150,85,38,133,61)(11,155,103,49,138,77,20,121,72)(12,162,114,50,145,88,21,128,64)(13,169,106,51,152,80,22,116,75)(14,157,98,52,140,91,23,123,67)(15,164,109,53,147,83,24,130,59)(16,171,101,54,135,94,25,118,70)(17,159,112,55,142,86,26,125,62)(18,166,104,56,149,78,27,132,73)(19,154,96,57,137,89,28,120,65)>;

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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171), (1,161,107,39,144,81,29,127,76)(2,168,99,40,151,92,30,115,68)(3,156,110,41,139,84,31,122,60)(4,163,102,42,146,95,32,129,71)(5,170,113,43,134,87,33,117,63)(6,158,105,44,141,79,34,124,74)(7,165,97,45,148,90,35,131,66)(8,153,108,46,136,82,36,119,58)(9,160,100,47,143,93,37,126,69)(10,167,111,48,150,85,38,133,61)(11,155,103,49,138,77,20,121,72)(12,162,114,50,145,88,21,128,64)(13,169,106,51,152,80,22,116,75)(14,157,98,52,140,91,23,123,67)(15,164,109,53,147,83,24,130,59)(16,171,101,54,135,94,25,118,70)(17,159,112,55,142,86,26,125,62)(18,166,104,56,149,78,27,132,73)(19,154,96,57,137,89,28,120,65) );

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),(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95),(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114),(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133),(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171)], [(1,161,107,39,144,81,29,127,76),(2,168,99,40,151,92,30,115,68),(3,156,110,41,139,84,31,122,60),(4,163,102,42,146,95,32,129,71),(5,170,113,43,134,87,33,117,63),(6,158,105,44,141,79,34,124,74),(7,165,97,45,148,90,35,131,66),(8,153,108,46,136,82,36,119,58),(9,160,100,47,143,93,37,126,69),(10,167,111,48,150,85,38,133,61),(11,155,103,49,138,77,20,121,72),(12,162,114,50,145,88,21,128,64),(13,169,106,51,152,80,22,116,75),(14,157,98,52,140,91,23,123,67),(15,164,109,53,147,83,24,130,59),(16,171,101,54,135,94,25,118,70),(17,159,112,55,142,86,26,125,62),(18,166,104,56,149,78,27,132,73),(19,154,96,57,137,89,28,120,65)])

C192C9 is a maximal subgroup of   C57.C6

Matrix representation of C192C9 in GL3(𝔽7) generated by

252
220
046
,
224
302
105
G:=sub<GL(3,GF(7))| [2,2,0,5,2,4,2,0,6],[2,3,1,2,0,0,4,2,5] >;

C192C9 in GAP, Magma, Sage, TeX

C_{19}\rtimes_2C_9
% in TeX

G:=Group("C19:2C9");
// GroupNames label

G:=SmallGroup(171,1);
// by ID

G=gap.SmallGroup(171,1);
# by ID

G:=PCGroup([3,-3,-3,-19,9,569]);
// Polycyclic

G:=Group<a,b|a^19=b^9=1,b*a*b^-1=a^11>;
// generators/relations

Export

Subgroup lattice of C192C9 in TeX
Character table of C192C9 in TeX

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