C9xD21 - GroupNames

G = C₉xD₂₁ order 378 = 2·3³·7

Direct product of C₉ and D₂₁

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

Aliases: C₉xD₂₁, C₆₃:₃S₃, C₂₁:₅C₁₈, C₃:(C₉xD₇), C₇:₃(S₃xC₉), (C₃xC₉):₁D₇, (C₃xC₆₃):₇C₂, (C₃xC₂₁).₉C₆, C₃.₄(C₃xD₂₁), C₂₁.₁₄(C₃xS₃), (C₃xD₂₁).₂C₃, C₃².₂(C₃xD₇), SmallGroup(378,37)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₁ — C₉xD₂₁

Chief series

C₁ — C₇ — C₂₁ — C₃xC₂₁ — C₃xC₆₃ — C₉xD₂₁

Lower central

C₂₁ — C₉xD₂₁

Upper central

C₁ — C₉

Generators and relations for C₉xD₂₁
G = < a,b,c | a⁹=b²¹=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 116 in 28 conjugacy classes, 15 normal (all characteristic)
Quotients: C₁, C₂, C₃, S₃, C₆, C₉, D₇, C₁₈, C₃xS₃, C₃xD₇, D₂₁, S₃xC₉, C₉xD₇, C₃xD₂₁, C₉xD₂₁

C₁

₂₁C₂

C₃

₂C₃

C₇

₇S₃

₂₁C₆

C₉

C₃²

₂C₉

₃D₇

C₂₁

₂C₂₁

₇C₃xS₃

₂₁C₁₈

C₃xC₉

D₂₁

₃C₃xD₇

C₆₃

C₃xC₂₁

₂C₆₃

₇S₃xC₉

C₃xD₂₁

₃C₉xD₇

C₃xC₆₃

C₉xD₂₁

Smallest permutation representation of C₉xD₂₁
►On 126 points

Generators in S₁₂₆

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

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

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

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

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

108 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	6A	6B	7A	7B	7C	9A	···	9F	9G	···	9L	18A	···	18F	21A	···	21X	63A	···	63BB
order	1	2	3	3	3	3	3	6	6	7	7	7	9	···	9	9	···	9	18	···	18	21	···	21	63	···	63
size	1	21	1	1	2	2	2	21	21	2	2	2	1	···	1	2	···	2	21	···	21	2	···	2	2	···	2

108 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2
type	+	+					+	+		+
image	C₁	C₂	C₃	C₆	C₉	C₁₈	S₃	D₇	C₃xS₃	D₂₁	C₃xD₇	S₃xC₉	C₉xD₇	C₃xD₂₁	C₉xD₂₁
kernel	C₉xD₂₁	C₃xC₆₃	C₃xD₂₁	C₃xC₂₁	D₂₁	C₂₁	C₆₃	C₃xC₉	C₂₁	C₉	C₃²	C₇	C₃	C₃	C₁
# reps	1	1	2	2	6	6	1	3	2	6	6	6	18	12	36

Matrix representation of C₉xD₂₁ ►in GL₂(F₁₂₇) generated by

52	0
0	52

94	0
0	50

0	50
94	0

G:=sub<GL(2,GF(127))| [52,0,0,52],[94,0,0,50],[0,94,50,0] >;

C₉xD₂₁ in GAP, Magma, Sage, TeX

C_9\times D_{21}

% in TeX

G:=Group("C9xD21");

// GroupNames label

G:=SmallGroup(378,37);

// by ID

G=gap.SmallGroup(378,37);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,36,723,8104]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₉xD₂₁ in TeX

G = C9xD21 order 378 = 2·33·7

Direct product of C9 and D21

G = C₉xD₂₁ order 378 = 2·3³·7

Direct product of C₉ and D₂₁