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G = C2×C7⋊C9order 126 = 2·32·7

Direct product of C2 and C7⋊C9

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

Aliases: C2×C7⋊C9, C14⋊C9, C72C18, C42.C3, C21.2C6, C6.(C7⋊C3), C3.(C2×C7⋊C3), SmallGroup(126,2)

Series: Derived Chief Lower central Upper central

C1C7 — C2×C7⋊C9
C1C7C21C7⋊C9 — C2×C7⋊C9
C7 — C2×C7⋊C9
C1C6

Generators and relations for C2×C7⋊C9
 G = < a,b,c | a2=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >

7C9
7C18

Character table of C2×C7⋊C9

 class 123A3B6A6B7A7B9A9B9C9D9E9F14A14B18A18B18C18D18E18F21A21B21C21D42A42B42C42D
 size 111111337777773377777733333333
ρ1111111111111111111111111111111    trivial
ρ21-111-1-111111111-1-1-1-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ31-111-1-111ζ32ζ3ζ3ζ32ζ3ζ32-1-1ζ65ζ6ζ6ζ6ζ65ζ651111-1-1-1-1    linear of order 6
ρ411111111ζ32ζ3ζ3ζ32ζ3ζ3211ζ3ζ32ζ32ζ32ζ3ζ311111111    linear of order 3
ρ511111111ζ3ζ32ζ32ζ3ζ32ζ311ζ32ζ3ζ3ζ3ζ32ζ3211111111    linear of order 3
ρ61-111-1-111ζ3ζ32ζ32ζ3ζ32ζ3-1-1ζ6ζ65ζ65ζ65ζ6ζ61111-1-1-1-1    linear of order 6
ρ71-1ζ32ζ3ζ6ζ6511ζ97ζ92ζ95ζ9ζ98ζ94-1-195949799892ζ32ζ3ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 18
ρ81-1ζ3ζ32ζ65ζ611ζ92ζ97ζ94ζ98ζ9ζ95-1-194959298997ζ3ζ32ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 18
ρ91-1ζ32ζ3ζ6ζ6511ζ94ζ95ζ98ζ97ζ92ζ9-1-198994979295ζ32ζ3ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 18
ρ1011ζ32ζ3ζ32ζ311ζ94ζ95ζ98ζ97ζ92ζ911ζ98ζ9ζ94ζ97ζ92ζ95ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ111-1ζ3ζ32ζ65ζ611ζ95ζ94ζ9ζ92ζ97ζ98-1-199895929794ζ3ζ32ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 18
ρ1211ζ3ζ32ζ3ζ3211ζ98ζ9ζ97ζ95ζ94ζ9211ζ97ζ92ζ98ζ95ζ94ζ9ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ131-1ζ3ζ32ζ65ζ611ζ98ζ9ζ97ζ95ζ94ζ92-1-197929895949ζ3ζ32ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 18
ρ1411ζ32ζ3ζ32ζ311ζ97ζ92ζ95ζ9ζ98ζ9411ζ95ζ94ζ97ζ9ζ98ζ92ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ151-1ζ32ζ3ζ6ζ6511ζ9ζ98ζ92ζ94ζ95ζ97-1-192979949598ζ32ζ3ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 18
ρ1611ζ3ζ32ζ3ζ3211ζ95ζ94ζ9ζ92ζ97ζ9811ζ9ζ98ζ95ζ92ζ97ζ94ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ1711ζ3ζ32ζ3ζ3211ζ92ζ97ζ94ζ98ζ9ζ9511ζ94ζ95ζ92ζ98ζ9ζ97ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 9
ρ1811ζ32ζ3ζ32ζ311ζ9ζ98ζ92ζ94ζ95ζ9711ζ92ζ97ζ9ζ94ζ95ζ98ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 9
ρ193-333-3-3-1+-7/2-1--7/20000001+-7/21--7/2000000-1+-7/2-1+-7/2-1--7/2-1--7/21+-7/21--7/21--7/21+-7/2    complex lifted from C2×C7⋊C3
ρ203-333-3-3-1--7/2-1+-7/20000001--7/21+-7/2000000-1--7/2-1--7/2-1+-7/2-1+-7/21--7/21+-7/21+-7/21--7/2    complex lifted from C2×C7⋊C3
ρ21333333-1--7/2-1+-7/2000000-1+-7/2-1--7/2000000-1--7/2-1--7/2-1+-7/2-1+-7/2-1+-7/2-1--7/2-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ22333333-1+-7/2-1--7/2000000-1--7/2-1+-7/2000000-1+-7/2-1+-7/2-1--7/2-1--7/2-1--7/2-1+-7/2-1+-7/2-1--7/2    complex lifted from C7⋊C3
ρ233-3-3+3-3/2-3-3-3/23-3-3/23+3-3/2-1--7/2-1+-7/20000001--7/21+-7/2000000ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ73ζ743ζ723ζ73ζ763ζ753ζ7332ζ7632ζ7532ζ7332ζ7432ζ7232ζ7    complex faithful, Schur index 3
ρ2433-3+3-3/2-3-3-3/2-3+3-3/2-3-3-3/2-1--7/2-1+-7/2000000-1+-7/2-1--7/2000000ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7    complex lifted from C7⋊C9, Schur index 3
ρ253-3-3-3-3/2-3+3-3/23+3-3/23-3-3/2-1--7/2-1+-7/20000001--7/21+-7/2000000ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ732ζ7432ζ7232ζ732ζ7632ζ7532ζ733ζ763ζ753ζ733ζ743ζ723ζ7    complex faithful, Schur index 3
ρ2633-3+3-3/2-3-3-3/2-3+3-3/2-3-3-3/2-1+-7/2-1--7/2000000-1--7/2-1+-7/2000000ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73    complex lifted from C7⋊C9, Schur index 3
ρ273-3-3-3-3/2-3+3-3/23+3-3/23-3-3/2-1+-7/2-1--7/20000001+-7/21--7/2000000ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ7332ζ7632ζ7532ζ7332ζ7432ζ7232ζ73ζ743ζ723ζ73ζ763ζ753ζ73    complex faithful, Schur index 3
ρ2833-3-3-3/2-3+3-3/2-3-3-3/2-3+3-3/2-1--7/2-1+-7/2000000-1+-7/2-1--7/2000000ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7    complex lifted from C7⋊C9, Schur index 3
ρ293-3-3+3-3/2-3-3-3/23-3-3/23+3-3/2-1+-7/2-1--7/20000001+-7/21--7/2000000ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ733ζ763ζ753ζ733ζ743ζ723ζ732ζ7432ζ7232ζ732ζ7632ζ7532ζ73    complex faithful, Schur index 3
ρ3033-3-3-3/2-3+3-3/2-3-3-3/2-3+3-3/2-1+-7/2-1--7/2000000-1--7/2-1+-7/2000000ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73    complex lifted from C7⋊C9, Schur index 3

Smallest permutation representation of C2×C7⋊C9
Regular action on 126 points
Generators in S126
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 46)(9 47)(10 88)(11 89)(12 90)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 116)(20 117)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 73)(35 74)(36 75)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 100)(62 101)(63 102)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)
(1 93 86 67 73 111 104)(2 74 94 112 87 105 68)(3 88 75 106 95 69 113)(4 96 89 70 76 114 107)(5 77 97 115 90 108 71)(6 82 78 100 98 72 116)(7 99 83 64 79 117 101)(8 80 91 109 84 102 65)(9 85 81 103 92 66 110)(10 36 58 122 42 25 50)(11 43 28 26 59 51 123)(12 60 44 52 29 124 27)(13 30 61 125 45 19 53)(14 37 31 20 62 54 126)(15 63 38 46 32 118 21)(16 33 55 119 39 22 47)(17 40 34 23 56 48 120)(18 57 41 49 35 121 24)
(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)

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

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

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

C2×C7⋊C9 is a maximal subgroup of   C7⋊C36  C18×C7⋊C3

Matrix representation of C2×C7⋊C9 in GL3(𝔽127) generated by

12600
01260
00126
,
126221
0221
126231
,
81315
427550
403498
G:=sub<GL(3,GF(127))| [126,0,0,0,126,0,0,0,126],[126,0,126,22,22,23,1,1,1],[81,42,40,31,75,34,5,50,98] >;

C2×C7⋊C9 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes C_9
% in TeX

G:=Group("C2xC7:C9");
// GroupNames label

G:=SmallGroup(126,2);
// by ID

G=gap.SmallGroup(126,2);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,29,295]);
// Polycyclic

G:=Group<a,b,c|a^2=b^7=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C2×C7⋊C9 in TeX
Character table of C2×C7⋊C9 in TeX

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