Copied to
clipboard

## G = D9×C21order 378 = 2·33·7

### Direct product of C21 and D9

Aliases: D9×C21, C93C42, C6318C6, (C3×C9)⋊2C14, (C3×C63)⋊4C2, C3.1(S3×C21), (C3×C21).5S3, C21.15(C3×S3), C32.2(S3×C7), SmallGroup(378,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9×C21
 Chief series C1 — C3 — C9 — C63 — C3×C63 — D9×C21
 Lower central C9 — D9×C21
 Upper central C1 — C21

Generators and relations for D9×C21
G = < a,b,c | a21=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D9×C21
On 126 points
Generators in S126
(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)
(1 112 65 8 119 72 15 126 79)(2 113 66 9 120 73 16 106 80)(3 114 67 10 121 74 17 107 81)(4 115 68 11 122 75 18 108 82)(5 116 69 12 123 76 19 109 83)(6 117 70 13 124 77 20 110 84)(7 118 71 14 125 78 21 111 64)(22 85 50 36 99 43 29 92 57)(23 86 51 37 100 44 30 93 58)(24 87 52 38 101 45 31 94 59)(25 88 53 39 102 46 32 95 60)(26 89 54 40 103 47 33 96 61)(27 90 55 41 104 48 34 97 62)(28 91 56 42 105 49 35 98 63)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(49 125)(50 126)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)

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

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

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

126 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 6A 6B 7A ··· 7F 9A ··· 9I 14A ··· 14F 21A ··· 21L 21M ··· 21AD 42A ··· 42L 63A ··· 63BB order 1 2 3 3 3 3 3 6 6 7 ··· 7 9 ··· 9 14 ··· 14 21 ··· 21 21 ··· 21 42 ··· 42 63 ··· 63 size 1 9 1 1 2 2 2 9 9 1 ··· 1 2 ··· 2 9 ··· 9 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + image C1 C2 C3 C6 C7 C14 C21 C42 S3 D9 C3×S3 S3×C7 C3×D9 C7×D9 S3×C21 D9×C21 kernel D9×C21 C3×C63 C7×D9 C63 C3×D9 C3×C9 D9 C9 C3×C21 C21 C21 C32 C7 C3 C3 C1 # reps 1 1 2 2 6 6 12 12 1 3 2 6 6 18 12 36

Matrix representation of D9×C21 in GL2(𝔽127) generated by

 100 0 0 100
,
 52 0 53 22
,
 105 46 53 22
G:=sub<GL(2,GF(127))| [100,0,0,100],[52,53,0,22],[105,53,46,22] >;

D9×C21 in GAP, Magma, Sage, TeX

D_9\times C_{21}
% in TeX

G:=Group("D9xC21");
// GroupNames label

G:=SmallGroup(378,32);
// by ID

G=gap.SmallGroup(378,32);
# by ID

G:=PCGroup([5,-2,-3,-7,-3,-3,4203,138,6304]);
// Polycyclic

G:=Group<a,b,c|a^21=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

׿
×
𝔽