C3xD63 - GroupNames

G = C₃xD₆₃ order 378 = 2·3³·7

Direct product of C₃ and D₆₃

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

Aliases: C₃xD₆₃, C₂₁:₂D₉, C₆₃:₁₇C₆, C₃².₂D₂₁, (C₃xC₉):₂D₇, C₇:₃(C₃xD₉), (C₃xC₆₃):₂C₂, C₉:₃(C₃xD₇), (C₃xC₂₁).₄S₃, C₃.₁(C₃xD₂₁), C₂₁.₁₁(C₃xS₃), SmallGroup(378,36)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₃ — C₃xD₆₃

Chief series

C₁ — C₃ — C₂₁ — C₆₃ — C₃xC₆₃ — C₃xD₆₃

Lower central

C₆₃ — C₃xD₆₃

Upper central

C₁ — C₃

Generators and relations for C₃xD₆₃
G = < a,b,c | a³=b⁶³=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 228 in 28 conjugacy classes, 14 normal (all characteristic)
Quotients: C₁, C₂, C₃, S₃, C₆, D₇, D₉, C₃xS₃, C₃xD₇, D₂₁, C₃xD₉, D₆₃, C₃xD₂₁, C₃xD₆₃

C₁

₆₃C₂

C₃

₂C₃

C₇

₂₁S₃

₆₃C₆

C₃²

C₉

₂C₉

₉D₇

C₂₁

₂C₂₁

₇D₉

₂₁C₃xS₃

C₃xC₉

₃D₂₁

₉C₃xD₇

C₃xC₂₁

C₆₃

₂C₆₃

₇C₃xD₉

D₆₃

₃C₃xD₂₁

C₃xC₆₃

C₃xD₆₃

Smallest permutation representation of C₃xD₆₃
►On 126 points

Generators in S₁₂₆

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

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

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

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

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

99 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	6A	6B	7A	7B	7C	9A	···	9I	21A	···	21X	63A	···	63BB
order	1	2	3	3	3	3	3	6	6	7	7	7	9	···	9	21	···	21	63	···	63
size	1	63	1	1	2	2	2	63	63	2	2	2	2	···	2	2	···	2	2	···	2

99 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+			+	+	+			+		+
image	C₁	C₂	C₃	C₆	S₃	D₇	D₉	C₃xS₃	C₃xD₇	D₂₁	C₃xD₉	D₆₃	C₃xD₂₁	C₃xD₆₃
kernel	C₃xD₆₃	C₃xC₆₃	D₆₃	C₆₃	C₃xC₂₁	C₃xC₉	C₂₁	C₂₁	C₉	C₃²	C₇	C₃	C₃	C₁
# reps	1	1	2	2	1	3	3	2	6	6	6	18	12	36

Matrix representation of C₃xD₆₃ ►in GL₂(F₁₂₇) generated by

19	0
0	19

113	0
0	9

0	1
1	0

G:=sub<GL(2,GF(127))| [19,0,0,19],[113,0,0,9],[0,1,1,0] >;

C₃xD₆₃ in GAP, Magma, Sage, TeX

C_3\times D_{63}

% in TeX

G:=Group("C3xD63");

// GroupNames label

G:=SmallGroup(378,36);

// by ID

G=gap.SmallGroup(378,36);

# by ID

G:=PCGroup([5,-2,-3,-3,-7,-3,2072,642,2163,6304]);

// Polycyclic

G:=Group<a,b,c|a^3=b^63=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

Export

Subgroup lattice of C₃xD₆₃ in TeX

G = C3xD63 order 378 = 2·33·7

Direct product of C3 and D63

G = C₃xD₆₃ order 378 = 2·3³·7

Direct product of C₃ and D₆₃