D9xC21 - GroupNames

G = D₉xC₂₁ order 378 = 2·3³·7

Direct product of C₂₁ and D₉

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

Aliases: D₉xC₂₁, C₉:₃C₄₂, C₆₃:₁₈C₆, (C₃xC₉):₂C₁₄, (C₃xC₆₃):₄C₂, C₃.₁(S₃xC₂₁), (C₃xC₂₁).₅S₃, C₂₁.₁₅(C₃xS₃), C₃².₂(S₃xC₇), SmallGroup(378,32)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₉ — D₉xC₂₁

Chief series

C₁ — C₃ — C₉ — C₆₃ — C₃xC₆₃ — D₉xC₂₁

Lower central

C₉ — D₉xC₂₁

Upper central

C₁ — C₂₁

Generators and relations for D₉xC₂₁
G = < a,b,c | a²¹=b⁹=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 72 in 28 conjugacy classes, 16 normal (all characteristic)
Quotients: C₁, C₂, C₃, S₃, C₆, C₇, C₁₄, D₉, C₃xS₃, C₂₁, S₃xC₇, C₄₂, C₃xD₉, C₇xD₉, S₃xC₂₁, D₉xC₂₁

C₁

₉C₂

C₃

₂C₃

C₇

₃S₃

₉C₆

C₃²

C₉

₂C₉

₉C₁₄

C₂₁

₂C₂₁

D₉

₃C₃xS₃

C₃xC₉

₃S₃xC₇

₉C₄₂

C₃xC₂₁

C₆₃

₂C₆₃

C₃xD₉

C₇xD₉

₃S₃xC₂₁

C₃xC₆₃

D₉xC₂₁

Smallest permutation representation of D₉xC₂₁
►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	C₃xS₃	S₃xC₇	C₃xD₉	C₇xD₉	S₃xC₂₁	D₉xC₂₁
kernel	D₉xC₂₁	C₃xC₆₃	C₇xD₉	C₆₃	C₃xD₉	C₃xC₉	D₉	C₉	C₃xC₂₁	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 D₉xC₂₁ ►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] >;

D₉xC₂₁ 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

Subgroup lattice of D₉xC₂₁ in TeX

G = D9xC21 order 378 = 2·33·7

Direct product of C21 and D9

G = D₉xC₂₁ order 378 = 2·3³·7

Direct product of C₂₁ and D₉