Copied to
clipboard

G = C57.C6order 342 = 2·32·19

The non-split extension by C57 of C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C57.C6, C193C18, D192C9, C192C9⋊C2, C3.(C19⋊C6), (C3×D19).C3, SmallGroup(342,1)

Series: Derived Chief Lower central Upper central

C1C19 — C57.C6
C1C19C57C192C9 — C57.C6
C19 — C57.C6
C1C3

Generators and relations for C57.C6
 G = < a,b | a57=1, b6=a19, bab-1=a31 >

19C2
19C6
19C9
19C18

Character table of C57.C6

 class 123A3B6A6B9A9B9C9D9E9F18A18B18C18D18E18F19A19B19C57A57B57C57D57E57F
 size 119111919191919191919191919191919666666666
ρ1111111111111111111111111111    trivial
ρ21-111-1-1111111-1-1-1-1-1-1111111111    linear of order 2
ρ3111111ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3111111111    linear of order 3
ρ4111111ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32111111111    linear of order 3
ρ51-111-1-1ζ32ζ3ζ3ζ32ζ32ζ3ζ65ζ6ζ6ζ6ζ65ζ65111111111    linear of order 6
ρ61-111-1-1ζ3ζ32ζ32ζ3ζ3ζ32ζ6ζ65ζ65ζ65ζ6ζ6111111111    linear of order 6
ρ711ζ32ζ3ζ32ζ3ζ97ζ92ζ95ζ94ζ9ζ98ζ95ζ94ζ97ζ9ζ98ζ92111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 9
ρ811ζ32ζ3ζ32ζ3ζ94ζ95ζ98ζ9ζ97ζ92ζ98ζ9ζ94ζ97ζ92ζ95111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 9
ρ91-1ζ3ζ32ζ65ζ6ζ98ζ9ζ97ζ92ζ95ζ9497929895949111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 18
ρ101-1ζ32ζ3ζ6ζ65ζ94ζ95ζ98ζ9ζ97ζ9298994979295111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 18
ρ1111ζ3ζ32ζ3ζ32ζ98ζ9ζ97ζ92ζ95ζ94ζ97ζ92ζ98ζ95ζ94ζ9111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 9
ρ1211ζ3ζ32ζ3ζ32ζ95ζ94ζ9ζ98ζ92ζ97ζ9ζ98ζ95ζ92ζ97ζ94111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 9
ρ131-1ζ3ζ32ζ65ζ6ζ92ζ97ζ94ζ95ζ98ζ994959298997111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 18
ρ141-1ζ3ζ32ζ65ζ6ζ95ζ94ζ9ζ98ζ92ζ9799895929794111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 18
ρ1511ζ32ζ3ζ32ζ3ζ9ζ98ζ92ζ97ζ94ζ95ζ92ζ97ζ9ζ94ζ95ζ98111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 9
ρ161-1ζ32ζ3ζ6ζ65ζ97ζ92ζ95ζ94ζ9ζ9895949799892111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 18
ρ171-1ζ32ζ3ζ6ζ65ζ9ζ98ζ92ζ97ζ94ζ9592979949598111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 18
ρ1811ζ3ζ32ζ3ζ32ζ92ζ97ζ94ζ95ζ98ζ9ζ94ζ95ζ92ζ98ζ9ζ97111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 9
ρ19606600000000000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192    orthogonal lifted from C19⋊C6
ρ20606600000000000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194    orthogonal lifted from C19⋊C6
ρ21606600000000000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719    orthogonal lifted from C19⋊C6
ρ2260-3-3-3-3+3-300000000000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ3ζ19173ζ19163ζ19143ζ1953ζ1933ζ192ζ3ζ19153ζ19133ζ19103ζ1993ζ1963ζ194ζ3ζ19183ζ19123ζ19113ζ1983ζ1973ζ19ζ32ζ191832ζ191232ζ191132ζ19832ζ19732ζ19ζ32ζ191732ζ191632ζ191432ζ19532ζ19332ζ192ζ32ζ191532ζ191332ζ191032ζ19932ζ19632ζ194    complex faithful
ρ2360-3-3-3-3+3-300000000000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ3ζ19153ζ19133ζ19103ζ1993ζ1963ζ194ζ3ζ19183ζ19123ζ19113ζ1983ζ1973ζ19ζ3ζ19173ζ19163ζ19143ζ1953ζ1933ζ192ζ32ζ191732ζ191632ζ191432ζ19532ζ19332ζ192ζ32ζ191532ζ191332ζ191032ζ19932ζ19632ζ194ζ32ζ191832ζ191232ζ191132ζ19832ζ19732ζ19    complex faithful
ρ2460-3-3-3-3+3-300000000000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ3ζ19183ζ19123ζ19113ζ1983ζ1973ζ19ζ3ζ19173ζ19163ζ19143ζ1953ζ1933ζ192ζ3ζ19153ζ19133ζ19103ζ1993ζ1963ζ194ζ32ζ191532ζ191332ζ191032ζ19932ζ19632ζ194ζ32ζ191832ζ191232ζ191132ζ19832ζ19732ζ19ζ32ζ191732ζ191632ζ191432ζ19532ζ19332ζ192    complex faithful
ρ2560-3+3-3-3-3-300000000000000ζ19181912191119819719ζ191719161914195193192ζ191519131910199196194ζ32ζ191532ζ191332ζ191032ζ19932ζ19632ζ194ζ32ζ191832ζ191232ζ191132ζ19832ζ19732ζ19ζ32ζ191732ζ191632ζ191432ζ19532ζ19332ζ192ζ3ζ19173ζ19163ζ19143ζ1953ζ1933ζ192ζ3ζ19153ζ19133ζ19103ζ1993ζ1963ζ194ζ3ζ19183ζ19123ζ19113ζ1983ζ1973ζ19    complex faithful
ρ2660-3+3-3-3-3-300000000000000ζ191519131910199196194ζ19181912191119819719ζ191719161914195193192ζ32ζ191732ζ191632ζ191432ζ19532ζ19332ζ192ζ32ζ191532ζ191332ζ191032ζ19932ζ19632ζ194ζ32ζ191832ζ191232ζ191132ζ19832ζ19732ζ19ζ3ζ19183ζ19123ζ19113ζ1983ζ1973ζ19ζ3ζ19173ζ19163ζ19143ζ1953ζ1933ζ192ζ3ζ19153ζ19133ζ19103ζ1993ζ1963ζ194    complex faithful
ρ2760-3+3-3-3-3-300000000000000ζ191719161914195193192ζ191519131910199196194ζ19181912191119819719ζ32ζ191832ζ191232ζ191132ζ19832ζ19732ζ19ζ32ζ191732ζ191632ζ191432ζ19532ζ19332ζ192ζ32ζ191532ζ191332ζ191032ζ19932ζ19632ζ194ζ3ζ19153ζ19133ζ19103ζ1993ζ1963ζ194ζ3ζ19183ζ19123ζ19113ζ1983ζ1973ζ19ζ3ζ19173ζ19163ζ19143ζ1953ζ1933ζ192    complex faithful

Smallest permutation representation of C57.C6
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 67 120 39 105 158 20 86 139)(2 113 127 19 97 132 21 75 146 38 59 151 40 94 165 57 78 170)(3 102 134 56 89 163 22 64 153 18 108 125 41 83 115 37 70 144)(4 91 141 36 81 137 23 110 160 55 100 156 42 72 122 17 62 118)(5 80 148 16 73 168 24 99 167 35 92 130 43 61 129 54 111 149)(6 69 155 53 65 142 25 88 117 15 84 161 44 107 136 34 103 123)(7 58 162 33 114 116 26 77 124 52 76 135 45 96 143 14 95 154)(8 104 169 13 106 147 27 66 131 32 68 166 46 85 150 51 87 128)(9 93 119 50 98 121 28 112 138 12 60 140 47 74 157 31 79 159)(10 82 126 30 90 152 29 101 145 49 109 171 48 63 164 11 71 133)

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

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

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

Matrix representation of C57.C6 in GL7(𝔽2053)

1855000000
01311454146614568631453
06005891329191013311321
0732132314637217311
020522052192260013202051
02146585415951455733
01320599119059913202052
,
276000000
010442981367125928278
061412229339331222614
027892812513672981044
0586204495336610371238
01543286666177317501286
001652142151514211652

G:=sub<GL(7,GF(2053))| [1855,0,0,0,0,0,0,0,131,600,732,2052,2,1320,0,1454,589,1323,2052,1465,599,0,1466,1329,1463,1922,854,1190,0,1456,1910,721,600,1595,599,0,863,1331,731,1320,1455,1320,0,1453,1321,1,2051,733,2052],[276,0,0,0,0,0,0,0,1044,614,278,586,1543,0,0,298,1222,928,2044,286,1652,0,1367,933,125,953,666,1421,0,125,933,1367,366,1773,515,0,928,1222,298,1037,1750,1421,0,278,614,1044,1238,1286,1652] >;

C57.C6 in GAP, Magma, Sage, TeX

C_{57}.C_6
% in TeX

G:=Group("C57.C6");
// GroupNames label

G:=SmallGroup(342,1);
// by ID

G=gap.SmallGroup(342,1);
# by ID

G:=PCGroup([4,-2,-3,-3,-19,29,5187,1015]);
// Polycyclic

G:=Group<a,b|a^57=1,b^6=a^19,b*a*b^-1=a^31>;
// generators/relations

Export

Subgroup lattice of C57.C6 in TeX
Character table of C57.C6 in TeX

׿
×
𝔽