Copied to
clipboard

## G = S3×C3×C21order 378 = 2·33·7

### Direct product of C3×C21 and S3

Aliases: S3×C3×C21, C331C14, C323C42, C3⋊(C3×C42), C217(C3×C6), (C3×C21)⋊18C6, (C32×C21)⋊5C2, SmallGroup(378,54)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C3×C21
 Chief series C1 — C3 — C21 — C3×C21 — C32×C21 — S3×C3×C21
 Lower central C3 — S3×C3×C21
 Upper central C1 — C3×C21

Generators and relations for S3×C3×C21
G = < a,b,c,d | a3=b21=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 104 in 64 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C7, C32, C32, C32, C14, C3×S3, C3×C6, C21, C21, C21, C33, S3×C7, C42, S3×C32, C3×C21, C3×C21, C3×C21, S3×C21, C3×C42, C32×C21, S3×C3×C21
Quotients: C1, C2, C3, S3, C6, C7, C32, C14, C3×S3, C3×C6, C21, S3×C7, C42, S3×C32, C3×C21, S3×C21, C3×C42, S3×C3×C21

Smallest permutation representation of S3×C3×C21
On 126 points
Generators in S126
(1 65 115)(2 66 116)(3 67 117)(4 68 118)(5 69 119)(6 70 120)(7 71 121)(8 72 122)(9 73 123)(10 74 124)(11 75 125)(12 76 126)(13 77 106)(14 78 107)(15 79 108)(16 80 109)(17 81 110)(18 82 111)(19 83 112)(20 84 113)(21 64 114)(22 58 88)(23 59 89)(24 60 90)(25 61 91)(26 62 92)(27 63 93)(28 43 94)(29 44 95)(30 45 96)(31 46 97)(32 47 98)(33 48 99)(34 49 100)(35 50 101)(36 51 102)(37 52 103)(38 53 104)(39 54 105)(40 55 85)(41 56 86)(42 57 87)
(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 79 122)(2 80 123)(3 81 124)(4 82 125)(5 83 126)(6 84 106)(7 64 107)(8 65 108)(9 66 109)(10 67 110)(11 68 111)(12 69 112)(13 70 113)(14 71 114)(15 72 115)(16 73 116)(17 74 117)(18 75 118)(19 76 119)(20 77 120)(21 78 121)(22 95 51)(23 96 52)(24 97 53)(25 98 54)(26 99 55)(27 100 56)(28 101 57)(29 102 58)(30 103 59)(31 104 60)(32 105 61)(33 85 62)(34 86 63)(35 87 43)(36 88 44)(37 89 45)(38 90 46)(39 91 47)(40 92 48)(41 93 49)(42 94 50)
(1 62)(2 63)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 106)(39 107)(40 108)(41 109)(42 110)(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)| (1,65,115)(2,66,116)(3,67,117)(4,68,118)(5,69,119)(6,70,120)(7,71,121)(8,72,122)(9,73,123)(10,74,124)(11,75,125)(12,76,126)(13,77,106)(14,78,107)(15,79,108)(16,80,109)(17,81,110)(18,82,111)(19,83,112)(20,84,113)(21,64,114)(22,58,88)(23,59,89)(24,60,90)(25,61,91)(26,62,92)(27,63,93)(28,43,94)(29,44,95)(30,45,96)(31,46,97)(32,47,98)(33,48,99)(34,49,100)(35,50,101)(36,51,102)(37,52,103)(38,53,104)(39,54,105)(40,55,85)(41,56,86)(42,57,87), (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), (1,79,122)(2,80,123)(3,81,124)(4,82,125)(5,83,126)(6,84,106)(7,64,107)(8,65,108)(9,66,109)(10,67,110)(11,68,111)(12,69,112)(13,70,113)(14,71,114)(15,72,115)(16,73,116)(17,74,117)(18,75,118)(19,76,119)(20,77,120)(21,78,121)(22,95,51)(23,96,52)(24,97,53)(25,98,54)(26,99,55)(27,100,56)(28,101,57)(29,102,58)(30,103,59)(31,104,60)(32,105,61)(33,85,62)(34,86,63)(35,87,43)(36,88,44)(37,89,45)(38,90,46)(39,91,47)(40,92,48)(41,93,49)(42,94,50), (1,62)(2,63)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,106)(39,107)(40,108)(41,109)(42,110)(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( (1,65,115)(2,66,116)(3,67,117)(4,68,118)(5,69,119)(6,70,120)(7,71,121)(8,72,122)(9,73,123)(10,74,124)(11,75,125)(12,76,126)(13,77,106)(14,78,107)(15,79,108)(16,80,109)(17,81,110)(18,82,111)(19,83,112)(20,84,113)(21,64,114)(22,58,88)(23,59,89)(24,60,90)(25,61,91)(26,62,92)(27,63,93)(28,43,94)(29,44,95)(30,45,96)(31,46,97)(32,47,98)(33,48,99)(34,49,100)(35,50,101)(36,51,102)(37,52,103)(38,53,104)(39,54,105)(40,55,85)(41,56,86)(42,57,87), (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), (1,79,122)(2,80,123)(3,81,124)(4,82,125)(5,83,126)(6,84,106)(7,64,107)(8,65,108)(9,66,109)(10,67,110)(11,68,111)(12,69,112)(13,70,113)(14,71,114)(15,72,115)(16,73,116)(17,74,117)(18,75,118)(19,76,119)(20,77,120)(21,78,121)(22,95,51)(23,96,52)(24,97,53)(25,98,54)(26,99,55)(27,100,56)(28,101,57)(29,102,58)(30,103,59)(31,104,60)(32,105,61)(33,85,62)(34,86,63)(35,87,43)(36,88,44)(37,89,45)(38,90,46)(39,91,47)(40,92,48)(41,93,49)(42,94,50), (1,62)(2,63)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,106)(39,107)(40,108)(41,109)(42,110)(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([[(1,65,115),(2,66,116),(3,67,117),(4,68,118),(5,69,119),(6,70,120),(7,71,121),(8,72,122),(9,73,123),(10,74,124),(11,75,125),(12,76,126),(13,77,106),(14,78,107),(15,79,108),(16,80,109),(17,81,110),(18,82,111),(19,83,112),(20,84,113),(21,64,114),(22,58,88),(23,59,89),(24,60,90),(25,61,91),(26,62,92),(27,63,93),(28,43,94),(29,44,95),(30,45,96),(31,46,97),(32,47,98),(33,48,99),(34,49,100),(35,50,101),(36,51,102),(37,52,103),(38,53,104),(39,54,105),(40,55,85),(41,56,86),(42,57,87)], [(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)], [(1,79,122),(2,80,123),(3,81,124),(4,82,125),(5,83,126),(6,84,106),(7,64,107),(8,65,108),(9,66,109),(10,67,110),(11,68,111),(12,69,112),(13,70,113),(14,71,114),(15,72,115),(16,73,116),(17,74,117),(18,75,118),(19,76,119),(20,77,120),(21,78,121),(22,95,51),(23,96,52),(24,97,53),(25,98,54),(26,99,55),(27,100,56),(28,101,57),(29,102,58),(30,103,59),(31,104,60),(32,105,61),(33,85,62),(34,86,63),(35,87,43),(36,88,44),(37,89,45),(38,90,46),(39,91,47),(40,92,48),(41,93,49),(42,94,50)], [(1,62),(2,63),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,106),(39,107),(40,108),(41,109),(42,110),(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)]])

189 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 7A ··· 7F 14A ··· 14F 21A ··· 21AV 21AW ··· 21CX 42A ··· 42AV order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 7 ··· 7 14 ··· 14 21 ··· 21 21 ··· 21 42 ··· 42 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

189 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C6 C7 C14 C21 C42 S3 C3×S3 S3×C7 S3×C21 kernel S3×C3×C21 C32×C21 S3×C21 C3×C21 S3×C32 C33 C3×S3 C32 C3×C21 C21 C32 C3 # reps 1 1 8 8 6 6 48 48 1 8 6 48

Matrix representation of S3×C3×C21 in GL4(𝔽43) generated by

 36 0 0 0 0 1 0 0 0 0 6 0 0 0 0 6
,
 23 0 0 0 0 6 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 6
,
 42 0 0 0 0 42 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(43))| [36,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[23,0,0,0,0,6,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,6],[42,0,0,0,0,42,0,0,0,0,0,1,0,0,1,0] >;

S3×C3×C21 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_{21}
% in TeX

G:=Group("S3xC3xC21");
// GroupNames label

G:=SmallGroup(378,54);
// by ID

G=gap.SmallGroup(378,54);
# by ID

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

G:=Group<a,b,c,d|a^3=b^21=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽