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G = C393C12order 468 = 22·32·13

1st semidirect product of C39 and C12 acting via C12/C2=C6

metacyclic, supersoluble, monomial, A-group

Aliases: C393C12, C78.1C6, Dic39⋊C3, C6.(C13⋊C6), C26.(C3×S3), C3⋊(C26.C6), C2.(D39⋊C3), C13⋊C32Dic3, C132(C3×Dic3), (C3×C13⋊C3)⋊3C4, (C2×C13⋊C3).S3, (C6×C13⋊C3).1C2, SmallGroup(468,21)

Series: Derived Chief Lower central Upper central

C1C39 — C393C12
C1C13C39C78C6×C13⋊C3 — C393C12
C39 — C393C12
C1C2

Generators and relations for C393C12
 G = < a,b | a39=b12=1, bab-1=a17 >

13C3
26C3
39C4
13C6
26C6
13C32
2C13⋊C3
13Dic3
39C12
13C3×C6
3Dic13
2C2×C13⋊C3
13C3×Dic3
3C26.C6

Character table of C393C12

 class 123A3B3C3D3E4A4B6A6B6C6D6E12A12B12C12D13A13B26A26B39A39B39C39D78A78B78C78D
 size 11213132626393921313262639393939666666666666
ρ1111111111111111111111111111111    trivial
ρ21111111-1-111111-1-1-1-1111111111111    linear of order 2
ρ3111ζ32ζ3ζ32ζ3-1-11ζ3ζ32ζ32ζ3ζ65ζ6ζ65ζ6111111111111    linear of order 6
ρ4111ζ32ζ3ζ32ζ3111ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32111111111111    linear of order 3
ρ5111ζ3ζ32ζ3ζ32-1-11ζ32ζ3ζ3ζ32ζ6ζ65ζ6ζ65111111111111    linear of order 6
ρ6111ζ3ζ32ζ3ζ32111ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3111111111111    linear of order 3
ρ71-111111i-i-1-1-1-1-1ii-i-i11-1-11111-1-1-1-1    linear of order 4
ρ81-111111-ii-1-1-1-1-1-i-iii11-1-11111-1-1-1-1    linear of order 4
ρ91-11ζ32ζ3ζ32ζ3-ii-1ζ65ζ6ζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ3211-1-11111-1-1-1-1    linear of order 12
ρ101-11ζ32ζ3ζ32ζ3i-i-1ζ65ζ6ζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ3211-1-11111-1-1-1-1    linear of order 12
ρ111-11ζ3ζ32ζ3ζ32-ii-1ζ6ζ65ζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ311-1-11111-1-1-1-1    linear of order 12
ρ121-11ζ3ζ32ζ3ζ32i-i-1ζ6ζ65ζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ311-1-11111-1-1-1-1    linear of order 12
ρ1322-122-1-100-122-1-100002222-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ142-2-122-1-1001-2-211000022-2-2-1-1-1-11111    symplectic lifted from Dic3, Schur index 2
ρ152-2-1-1--3-1+-3ζ6ζ650011--31+-3ζ32ζ3000022-2-2-1-1-1-11111    complex lifted from C3×Dic3
ρ1622-1-1+-3-1--3ζ65ζ600-1-1--3-1+-3ζ65ζ600002222-1-1-1-1-1-1-1-1    complex lifted from C3×S3
ρ172-2-1-1+-3-1--3ζ65ζ60011+-31--3ζ3ζ32000022-2-2-1-1-1-11111    complex lifted from C3×Dic3
ρ1822-1-1--3-1+-3ζ6ζ6500-1-1+-3-1--3ζ6ζ6500002222-1-1-1-1-1-1-1-1    complex lifted from C3×S3
ρ19666000000600000000-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2    orthogonal lifted from C13⋊C6
ρ2066-3000000-300000000-1+13/2-1-13/2-1+13/2-1-13/2ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132    orthogonal lifted from D39⋊C3
ρ21666000000600000000-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2    orthogonal lifted from C13⋊C6
ρ2266-3000000-300000000-1-13/2-1+13/2-1-13/2-1+13/23ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313913313    orthogonal lifted from D39⋊C3
ρ2366-3000000-300000000-1-13/2-1+13/2-1-13/2-1+13/2ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134    orthogonal lifted from D39⋊C3
ρ2466-3000000-300000000-1+13/2-1-13/2-1+13/2-1-13/232ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ1321311138137    orthogonal lifted from D39⋊C3
ρ256-6-3000000300000000-1+13/2-1-13/21-13/21+13/2ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ13213613513232ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313913313ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ1321311138137ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13131213101343ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132    symplectic faithful, Schur index 2
ρ266-66000000-600000000-1-13/2-1+13/21+13/21-13/2-1-13/2-1+13/2-1-13/2-1+13/21+13/21-13/21+13/21-13/2    symplectic lifted from C26.C6, Schur index 2
ρ276-66000000-600000000-1+13/2-1-13/21-13/21+13/2-1+13/2-1-13/2-1+13/2-1-13/21-13/21+13/21-13/21+13/2    symplectic lifted from C26.C6, Schur index 2
ρ286-6-3000000300000000-1+13/2-1-13/21-13/21+13/232ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ1321311138137ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13131213101343ζ13113ζ1383ζ1373ζ1363ζ1353ζ13213613513232ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313913313ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ1321311138137    symplectic faithful, Schur index 2
ρ296-6-3000000300000000-1-13/2-1+13/21+13/21-13/2ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13139133133ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ13131213101343ζ13113ζ1383ζ1373ζ1363ζ1353ζ13213613513232ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313913313ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ1321311138137ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134    symplectic faithful, Schur index 2
ρ306-6-3000000300000000-1-13/2-1+13/21+13/21-13/23ζ13113ζ1383ζ1373ζ1363ζ1353ζ132131113813732ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313121310134ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ132136135132ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313913313ζ3ζ13113ζ1383ζ1373ζ1363ζ1353ζ1321311138137ζ32ζ131232ζ131032ζ13932ζ13432ζ13332ζ13131213101343ζ13113ζ1383ζ1373ζ1363ζ1353ζ13213613513232ζ131232ζ131032ζ13932ζ13432ζ13332ζ1313913313    symplectic faithful, Schur index 2

Smallest permutation representation of C393C12
On 156 points
Generators in S156
(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)
(1 133 52 96)(2 156 74 95 17 150 53 80 23 132 68 113)(3 140 57 94 33 128 54 103 6 131 45 91)(4 124 40 93 10 145 55 87 28 130 61 108)(5 147 62 92 26 123 56 110 11 129 77 86)(7 154 67 90 19 118 58 117 16 127 70 81)(8 138 50 89 35 135 59 101 38 126 47 98)(9 122 72 88 12 152 60 85 21 125 63 115)(13 136 43 84 37 142 64 99 31 121 49 105)(14 120 65 83)(15 143 48 82 30 137 66 106 36 119 42 100)(18 134 75 79 39 149 69 97 24 155 51 112)(20 141 41 116 32 144 71 104 29 153 44 107)(22 148 46 114 25 139 73 111 34 151 76 102)(27 146 78 109)

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

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

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

Matrix representation of C393C12 in GL8(𝔽157)

13052000000
6826000000
0070138696913870
008720155898688
0069691687068
00891559015589156
00100000
00010000
,
5099000000
0107000000
0010148113356935
001133534798835
001467384015492
00969315049140147
004412244194444
003630149195511

G:=sub<GL(8,GF(157))| [130,68,0,0,0,0,0,0,52,26,0,0,0,0,0,0,0,0,70,87,69,89,1,0,0,0,138,20,69,155,0,1,0,0,69,155,1,90,0,0,0,0,69,89,68,155,0,0,0,0,138,86,70,89,0,0,0,0,70,88,68,156,0,0],[50,0,0,0,0,0,0,0,99,107,0,0,0,0,0,0,0,0,10,113,146,96,44,36,0,0,148,35,73,93,122,30,0,0,113,34,8,150,44,149,0,0,35,79,40,49,19,19,0,0,69,88,154,140,44,55,0,0,35,35,92,147,44,11] >;

C393C12 in GAP, Magma, Sage, TeX

C_{39}\rtimes_3C_{12}
% in TeX

G:=Group("C39:3C12");
// GroupNames label

G:=SmallGroup(468,21);
// by ID

G=gap.SmallGroup(468,21);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-13,30,483,10804,1359]);
// Polycyclic

G:=Group<a,b|a^39=b^12=1,b*a*b^-1=a^17>;
// generators/relations

Export

Subgroup lattice of C393C12 in TeX
Character table of C393C12 in TeX

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