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G = C2×C7⋊C18order 252 = 22·32·7

Direct product of C2 and C7⋊C18

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C2×C7⋊C18, D14⋊C9, C14⋊C18, D7⋊C18, C6.3F7, C42.3C6, C7⋊(C2×C18), C7⋊C9⋊C22, C3.(C2×F7), C21.(C2×C6), (C3×D7).C6, (C6×D7).C3, (C2×C7⋊C9)⋊C2, SmallGroup(252,7)

Series: Derived Chief Lower central Upper central

C1C7 — C2×C7⋊C18
C1C7C21C7⋊C9C7⋊C18 — C2×C7⋊C18
C7 — C2×C7⋊C18
C1C6

Generators and relations for C2×C7⋊C18
 G = < a,b,c | a2=b7=c18=1, ab=ba, ac=ca, cbc-1=b3 >

7C2
7C2
7C22
7C6
7C6
7C9
7C2×C6
7C18
7C18
7C18
7C2×C18

Smallest permutation representation of C2×C7⋊C18
On 126 points
Generators in S126
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 91)(34 92)(35 93)(36 94)(37 120)(38 121)(39 122)(40 123)(41 124)(42 125)(43 126)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 86)(56 87)(57 88)(58 89)(59 90)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)
(1 74 47 106 21 121 70)(2 107 71 48 122 75 22)(3 49 23 72 76 108 123)(4 55 124 24 91 50 77)(5 25 78 125 51 56 92)(6 126 93 79 57 26 52)(7 80 53 94 27 109 58)(8 95 59 54 110 81 28)(9 37 29 60 82 96 111)(10 61 112 30 97 38 83)(11 31 84 113 39 62 98)(12 114 99 85 63 32 40)(13 86 41 100 33 115 64)(14 101 65 42 116 87 34)(15 43 35 66 88 102 117)(16 67 118 36 103 44 89)(17 19 90 119 45 68 104)(18 120 105 73 69 20 46)
(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)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,91)(34,92)(35,93)(36,94)(37,120)(38,121)(39,122)(40,123)(41,124)(42,125)(43,126)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,86)(56,87)(57,88)(58,89)(59,90)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85), (1,74,47,106,21,121,70)(2,107,71,48,122,75,22)(3,49,23,72,76,108,123)(4,55,124,24,91,50,77)(5,25,78,125,51,56,92)(6,126,93,79,57,26,52)(7,80,53,94,27,109,58)(8,95,59,54,110,81,28)(9,37,29,60,82,96,111)(10,61,112,30,97,38,83)(11,31,84,113,39,62,98)(12,114,99,85,63,32,40)(13,86,41,100,33,115,64)(14,101,65,42,116,87,34)(15,43,35,66,88,102,117)(16,67,118,36,103,44,89)(17,19,90,119,45,68,104)(18,120,105,73,69,20,46), (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( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,91)(34,92)(35,93)(36,94)(37,120)(38,121)(39,122)(40,123)(41,124)(42,125)(43,126)(44,109)(45,110)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,86)(56,87)(57,88)(58,89)(59,90)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85), (1,74,47,106,21,121,70)(2,107,71,48,122,75,22)(3,49,23,72,76,108,123)(4,55,124,24,91,50,77)(5,25,78,125,51,56,92)(6,126,93,79,57,26,52)(7,80,53,94,27,109,58)(8,95,59,54,110,81,28)(9,37,29,60,82,96,111)(10,61,112,30,97,38,83)(11,31,84,113,39,62,98)(12,114,99,85,63,32,40)(13,86,41,100,33,115,64)(14,101,65,42,116,87,34)(15,43,35,66,88,102,117)(16,67,118,36,103,44,89)(17,19,90,119,45,68,104)(18,120,105,73,69,20,46), (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([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,91),(34,92),(35,93),(36,94),(37,120),(38,121),(39,122),(40,123),(41,124),(42,125),(43,126),(44,109),(45,110),(46,111),(47,112),(48,113),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,86),(56,87),(57,88),(58,89),(59,90),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85)], [(1,74,47,106,21,121,70),(2,107,71,48,122,75,22),(3,49,23,72,76,108,123),(4,55,124,24,91,50,77),(5,25,78,125,51,56,92),(6,126,93,79,57,26,52),(7,80,53,94,27,109,58),(8,95,59,54,110,81,28),(9,37,29,60,82,96,111),(10,61,112,30,97,38,83),(11,31,84,113,39,62,98),(12,114,99,85,63,32,40),(13,86,41,100,33,115,64),(14,101,65,42,116,87,34),(15,43,35,66,88,102,117),(16,67,118,36,103,44,89),(17,19,90,119,45,68,104),(18,120,105,73,69,20,46)], [(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)]])

42 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F 7 9A···9F 14 18A···18R21A21B42A42B
order12223366666679···91418···1821214242
size11771111777767···767···76666

42 irreducible representations

dim1111111116666
type+++++
imageC1C2C2C3C6C6C9C18C18F7C2×F7C7⋊C18C2×C7⋊C18
kernelC2×C7⋊C18C7⋊C18C2×C7⋊C9C6×D7C3×D7C42D14D7C14C6C3C2C1
# reps12124261261122

Matrix representation of C2×C7⋊C18 in GL8(𝔽127)

10000000
0126000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
0000000126
0010000126
0001000126
0000100126
0000010126
0000001126
,
590000000
01000000
0031246737090
0038704012430
0002737408790
0090874037270
0030124400873
0090037671243

G:=sub<GL(8,GF(127))| [1,0,0,0,0,0,0,0,0,126,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,126,126,126,126,126,126],[59,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,3,0,90,30,90,0,0,124,87,27,87,124,0,0,0,67,0,37,40,40,37,0,0,37,40,40,37,0,67,0,0,0,124,87,27,87,124,0,0,90,30,90,0,3,3] >;

C2×C7⋊C18 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes C_{18}
% in TeX

G:=Group("C2xC7:C18");
// GroupNames label

G:=SmallGroup(252,7);
// by ID

G=gap.SmallGroup(252,7);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,57,5404,914]);
// Polycyclic

G:=Group<a,b,c|a^2=b^7=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C2×C7⋊C18 in TeX

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