S3xC63 - GroupNames

G = S₃xC₆₃ order 378 = 2·3³·7

Direct product of C₆₃ and S₃

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

Aliases: S₃xC₆₃, C₃:C₁₂₆, C₂₁:₇C₁₈, C₃².₂C₄₂, (C₃xC₉):₁C₁₄, (C₃xC₆₃):₁C₂, (C₃xS₃).C₂₁, C₃.₄(S₃xC₂₁), (S₃xC₂₁).₂C₃, C₂₁.₁₈(C₃xS₃), (C₃xC₂₁).₁₁C₆, SmallGroup(378,33)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — S₃xC₆₃

Chief series

C₁ — C₃ — C₃² — C₃xC₂₁ — C₃xC₆₃ — S₃xC₆₃

Lower central

C₃ — S₃xC₆₃

Upper central

C₁ — C₆₃

Generators and relations for S₃xC₆₃
G = < a,b,c | a⁶³=b³=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 44 in 28 conjugacy classes, 18 normal (all characteristic)
Quotients: C₁, C₂, C₃, S₃, C₆, C₇, C₉, C₁₄, C₁₈, C₃xS₃, C₂₁, S₃xC₇, C₄₂, S₃xC₉, C₆₃, C₁₂₆, S₃xC₂₁, S₃xC₆₃

C₁

₃C₂

C₃

₂C₃

C₇

S₃

₃C₆

C₉

C₃²

₂C₉

₃C₁₄

C₂₁

₂C₂₁

C₃xS₃

₃C₁₈

C₃xC₉

S₃xC₇

₃C₄₂

C₆₃

C₃xC₂₁

₂C₆₃

S₃xC₉

S₃xC₂₁

₃C₁₂₆

C₃xC₆₃

S₃xC₆₃

Smallest permutation representation of S₃xC₆₃
►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 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)

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

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

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

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

189 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	6A	6B	7A	···	7F	9A	···	9F	9G	···	9L	14A	···	14F	18A	···	18F	21A	···	21L	21M	···	21AD	42A	···	42L	63A	···	63AJ	63AK	···	63BT	126A	···	126AJ
order	1	2	3	3	3	3	3	6	6	7	···	7	9	···	9	9	···	9	14	···	14	18	···	18	21	···	21	21	···	21	42	···	42	63	···	63	63	···	63	126	···	126
size	1	3	1	1	2	2	2	3	3	1	···	1	1	···	1	2	···	2	3	···	3	3	···	3	1	···	1	2	···	2	3	···	3	1	···	1	2	···	2	3	···	3

189 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+											+
image	C₁	C₂	C₃	C₆	C₇	C₉	C₁₄	C₁₈	C₂₁	C₄₂	C₆₃	C₁₂₆	S₃	C₃xS₃	S₃xC₇	S₃xC₉	S₃xC₂₁	S₃xC₆₃
kernel	S₃xC₆₃	C₃xC₆₃	S₃xC₂₁	C₃xC₂₁	S₃xC₉	S₃xC₇	C₃xC₉	C₂₁	C₃xS₃	C₃²	S₃	C₃	C₆₃	C₂₁	C₉	C₇	C₃	C₁
# reps	1	1	2	2	6	6	6	6	12	12	36	36	1	2	6	6	12	36

Matrix representation of S₃xC₆₃ ►in GL₂(F₁₂₇) generated by

72	0
0	72

107	0
0	19

0	1
1	0

G:=sub<GL(2,GF(127))| [72,0,0,72],[107,0,0,19],[0,1,1,0] >;

S₃xC₆₃ in GAP, Magma, Sage, TeX

S_3\times C_{63}

% in TeX

G:=Group("S3xC63");

// GroupNames label

G:=SmallGroup(378,33);

// by ID

G=gap.SmallGroup(378,33);

# by ID

G:=PCGroup([5,-2,-3,-7,-3,-3,216,6304]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of S₃xC₆₃ in TeX

G = S3xC63 order 378 = 2·33·7

Direct product of C63 and S3

G = S₃xC₆₃ order 378 = 2·3³·7

Direct product of C₆₃ and S₃