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G = C74.C6order 444 = 22·3·37

The non-split extension by C74 of C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C74.C6, C372C12, Dic37⋊C3, C37⋊C32C4, C2.(C37⋊C6), (C2×C37⋊C3).C2, SmallGroup(444,1)

Series: Derived Chief Lower central Upper central

C1C37 — C74.C6
C1C37C74C2×C37⋊C3 — C74.C6
C37 — C74.C6
C1C2

Generators and relations for C74.C6
 G = < a,b | a74=1, b6=a37, bab-1=a11 >

37C3
37C4
37C6
37C12

Character table of C74.C6

 class 123A3B4A4B6A6B12A12B12C12D37A37B37C37D37E37F74A74B74C74D74E74F
 size 1137373737373737373737666666666666
ρ1111111111111111111111111    trivial
ρ21111-1-111-1-1-1-1111111111111    linear of order 2
ρ311ζ3ζ3211ζ32ζ3ζ3ζ32ζ3ζ32111111111111    linear of order 3
ρ411ζ3ζ32-1-1ζ32ζ3ζ65ζ6ζ65ζ6111111111111    linear of order 6
ρ511ζ32ζ3-1-1ζ3ζ32ζ6ζ65ζ6ζ65111111111111    linear of order 6
ρ611ζ32ζ311ζ3ζ32ζ32ζ3ζ32ζ3111111111111    linear of order 3
ρ71-111-ii-1-1ii-i-i111111-1-1-1-1-1-1    linear of order 4
ρ81-111i-i-1-1-i-iii111111-1-1-1-1-1-1    linear of order 4
ρ91-1ζ3ζ32i-iζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32111111-1-1-1-1-1-1    linear of order 12
ρ101-1ζ32ζ3i-iζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3111111-1-1-1-1-1-1    linear of order 12
ρ111-1ζ3ζ32-iiζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32111111-1-1-1-1-1-1    linear of order 12
ρ121-1ζ32ζ3-iiζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3111111-1-1-1-1-1-1    linear of order 12
ρ13660000000000ζ3731372937233714378376ζ37283725372137163712379ζ37323724371937183713375ζ373437333730377374373ζ3736372737263711371037ζ37353722372037173715372ζ3736372737263711371037ζ37353722372037173715372ζ3731372937233714378376ζ37283725372137163712379ζ37323724371937183713375ζ373437333730377374373    orthogonal lifted from C37⋊C6
ρ14660000000000ζ373437333730377374373ζ3731372937233714378376ζ37283725372137163712379ζ37353722372037173715372ζ37323724371937183713375ζ3736372737263711371037ζ37323724371937183713375ζ3736372737263711371037ζ373437333730377374373ζ3731372937233714378376ζ37283725372137163712379ζ37353722372037173715372    orthogonal lifted from C37⋊C6
ρ15660000000000ζ37353722372037173715372ζ373437333730377374373ζ3731372937233714378376ζ3736372737263711371037ζ37283725372137163712379ζ37323724371937183713375ζ37283725372137163712379ζ37323724371937183713375ζ37353722372037173715372ζ373437333730377374373ζ3731372937233714378376ζ3736372737263711371037    orthogonal lifted from C37⋊C6
ρ16660000000000ζ3736372737263711371037ζ37353722372037173715372ζ373437333730377374373ζ37323724371937183713375ζ3731372937233714378376ζ37283725372137163712379ζ3731372937233714378376ζ37283725372137163712379ζ3736372737263711371037ζ37353722372037173715372ζ373437333730377374373ζ37323724371937183713375    orthogonal lifted from C37⋊C6
ρ17660000000000ζ37283725372137163712379ζ37323724371937183713375ζ3736372737263711371037ζ3731372937233714378376ζ37353722372037173715372ζ373437333730377374373ζ37353722372037173715372ζ373437333730377374373ζ37283725372137163712379ζ37323724371937183713375ζ3736372737263711371037ζ3731372937233714378376    orthogonal lifted from C37⋊C6
ρ18660000000000ζ37323724371937183713375ζ3736372737263711371037ζ37353722372037173715372ζ37283725372137163712379ζ373437333730377374373ζ3731372937233714378376ζ373437333730377374373ζ3731372937233714378376ζ37323724371937183713375ζ3736372737263711371037ζ37353722372037173715372ζ37283725372137163712379    orthogonal lifted from C37⋊C6
ρ196-60000000000ζ373437333730377374373ζ3731372937233714378376ζ37283725372137163712379ζ37353722372037173715372ζ37323724371937183713375ζ373637273726371137103737323724371937183713375373637273726371137103737343733373037737437337313729372337143783763728372537213716371237937353722372037173715372    symplectic faithful, Schur index 2
ρ206-60000000000ζ37353722372037173715372ζ373437333730377374373ζ3731372937233714378376ζ3736372737263711371037ζ37283725372137163712379ζ3732372437193718371337537283725372137163712379373237243719371837133753735372237203717371537237343733373037737437337313729372337143783763736372737263711371037    symplectic faithful, Schur index 2
ρ216-60000000000ζ37283725372137163712379ζ37323724371937183713375ζ3736372737263711371037ζ3731372937233714378376ζ37353722372037173715372ζ37343733373037737437337353722372037173715372373437333730377374373372837253721371637123793732372437193718371337537363727372637113710373731372937233714378376    symplectic faithful, Schur index 2
ρ226-60000000000ζ3731372937233714378376ζ37283725372137163712379ζ37323724371937183713375ζ373437333730377374373ζ3736372737263711371037ζ3735372237203717371537237363727372637113710373735372237203717371537237313729372337143783763728372537213716371237937323724371937183713375373437333730377374373    symplectic faithful, Schur index 2
ρ236-60000000000ζ37323724371937183713375ζ3736372737263711371037ζ37353722372037173715372ζ37283725372137163712379ζ373437333730377374373ζ373137293723371437837637343733373037737437337313729372337143783763732372437193718371337537363727372637113710373735372237203717371537237283725372137163712379    symplectic faithful, Schur index 2
ρ246-60000000000ζ3736372737263711371037ζ37353722372037173715372ζ373437333730377374373ζ37323724371937183713375ζ3731372937233714378376ζ3728372537213716371237937313729372337143783763728372537213716371237937363727372637113710373735372237203717371537237343733373037737437337323724371937183713375    symplectic faithful, Schur index 2

Smallest permutation representation of C74.C6
On 148 points
Generators in S148
(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)
(1 81 38 118)(2 108 27 117 48 92 39 145 64 80 11 129)(3 135 16 116 21 103 40 98 53 79 58 140)(4 88 5 115 68 114 41 125 42 78 31 77)(6 142 57 113 14 136 43 105 20 76 51 99)(7 95 46 112 61 147 44 132 9 75 24 110)(8 122 35 111 34 84 45 85 72 148 71 121)(10 102 13 109 54 106 47 139 50 146 17 143)(12 82 65 107 74 128 49 119 28 144 37 91)(15 89 32 104 67 87 52 126 69 141 30 124)(18 96 73 101 60 120 55 133 36 138 23 83)(19 123 62 100 33 131 56 86 25 137 70 94)(22 130 29 97 26 90 59 93 66 134 63 127)

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

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

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

Matrix representation of C74.C6 in GL7(𝔽1777)

1776000000
0791182907158010371141
06362058481345469451
01326149335335314931326
04514691345848205636
0114110371580907182791
09861596146616221336637
,
1200000000
0671224902732984947
015093453911100501086
096415902763451488765
0176548511178542241619
0145292817268661388756
016714798091412170020

G:=sub<GL(7,GF(1777))| [1776,0,0,0,0,0,0,0,791,636,1326,451,1141,986,0,182,205,1493,469,1037,1596,0,907,848,353,1345,1580,1466,0,1580,1345,353,848,907,1622,0,1037,469,1493,205,182,1336,0,1141,451,1326,636,791,637],[1200,0,0,0,0,0,0,0,671,1509,964,1765,1452,167,0,224,345,1590,485,928,1479,0,902,391,276,1117,1726,809,0,732,1100,345,854,866,1412,0,984,50,1488,224,1388,1700,0,947,1086,765,1619,756,20] >;

C74.C6 in GAP, Magma, Sage, TeX

C_{74}.C_6
% in TeX

G:=Group("C74.C6");
// GroupNames label

G:=SmallGroup(444,1);
// by ID

G=gap.SmallGroup(444,1);
# by ID

G:=PCGroup([4,-2,-3,-2,-37,24,6915,2503]);
// Polycyclic

G:=Group<a,b|a^74=1,b^6=a^37,b*a*b^-1=a^11>;
// generators/relations

Export

Subgroup lattice of C74.C6 in TeX
Character table of C74.C6 in TeX

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