direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×C21, C33⋊2C14, C32⋊4C42, C3⋊(S3×C21), C21⋊7(C3×S3), (C3×C21)⋊8S3, (C3×C21)⋊19C6, C32⋊3(S3×C7), (C32×C21)⋊6C2, SmallGroup(378,56)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3×C21 |
Generators and relations for C3⋊S3×C21
G = < a,b,c,d | a21=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 144 in 64 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C7, C32, C32, C32, C14, C3×S3, C3⋊S3, C21, C21, C21, C33, S3×C7, C42, C3×C3⋊S3, C3×C21, C3×C21, C3×C21, S3×C21, C7×C3⋊S3, C32×C21, C3⋊S3×C21
Quotients: C1, C2, C3, S3, C6, C7, C14, C3×S3, C3⋊S3, C21, S3×C7, C42, C3×C3⋊S3, S3×C21, C7×C3⋊S3, C3⋊S3×C21
►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 75 34)(2 76 35)(3 77 36)(4 78 37)(5 79 38)(6 80 39)(7 81 40)(8 82 41)(9 83 42)(10 84 22)(11 64 23)(12 65 24)(13 66 25)(14 67 26)(15 68 27)(16 69 28)(17 70 29)(18 71 30)(19 72 31)(20 73 32)(21 74 33)(43 122 88)(44 123 89)(45 124 90)(46 125 91)(47 126 92)(48 106 93)(49 107 94)(50 108 95)(51 109 96)(52 110 97)(53 111 98)(54 112 99)(55 113 100)(56 114 101)(57 115 102)(58 116 103)(59 117 104)(60 118 105)(61 119 85)(62 120 86)(63 121 87)
(1 41 68)(2 42 69)(3 22 70)(4 23 71)(5 24 72)(6 25 73)(7 26 74)(8 27 75)(9 28 76)(10 29 77)(11 30 78)(12 31 79)(13 32 80)(14 33 81)(15 34 82)(16 35 83)(17 36 84)(18 37 64)(19 38 65)(20 39 66)(21 40 67)(43 102 108)(44 103 109)(45 104 110)(46 105 111)(47 85 112)(48 86 113)(49 87 114)(50 88 115)(51 89 116)(52 90 117)(53 91 118)(54 92 119)(55 93 120)(56 94 121)(57 95 122)(58 96 123)(59 97 124)(60 98 125)(61 99 126)(62 100 106)(63 101 107)
(1 60)(2 61)(3 62)(4 63)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)
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), (1,75,34)(2,76,35)(3,77,36)(4,78,37)(5,79,38)(6,80,39)(7,81,40)(8,82,41)(9,83,42)(10,84,22)(11,64,23)(12,65,24)(13,66,25)(14,67,26)(15,68,27)(16,69,28)(17,70,29)(18,71,30)(19,72,31)(20,73,32)(21,74,33)(43,122,88)(44,123,89)(45,124,90)(46,125,91)(47,126,92)(48,106,93)(49,107,94)(50,108,95)(51,109,96)(52,110,97)(53,111,98)(54,112,99)(55,113,100)(56,114,101)(57,115,102)(58,116,103)(59,117,104)(60,118,105)(61,119,85)(62,120,86)(63,121,87), (1,41,68)(2,42,69)(3,22,70)(4,23,71)(5,24,72)(6,25,73)(7,26,74)(8,27,75)(9,28,76)(10,29,77)(11,30,78)(12,31,79)(13,32,80)(14,33,81)(15,34,82)(16,35,83)(17,36,84)(18,37,64)(19,38,65)(20,39,66)(21,40,67)(43,102,108)(44,103,109)(45,104,110)(46,105,111)(47,85,112)(48,86,113)(49,87,114)(50,88,115)(51,89,116)(52,90,117)(53,91,118)(54,92,119)(55,93,120)(56,94,121)(57,95,122)(58,96,123)(59,97,124)(60,98,125)(61,99,126)(62,100,106)(63,101,107), (1,60)(2,61)(3,62)(4,63)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93)>;
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), (1,75,34)(2,76,35)(3,77,36)(4,78,37)(5,79,38)(6,80,39)(7,81,40)(8,82,41)(9,83,42)(10,84,22)(11,64,23)(12,65,24)(13,66,25)(14,67,26)(15,68,27)(16,69,28)(17,70,29)(18,71,30)(19,72,31)(20,73,32)(21,74,33)(43,122,88)(44,123,89)(45,124,90)(46,125,91)(47,126,92)(48,106,93)(49,107,94)(50,108,95)(51,109,96)(52,110,97)(53,111,98)(54,112,99)(55,113,100)(56,114,101)(57,115,102)(58,116,103)(59,117,104)(60,118,105)(61,119,85)(62,120,86)(63,121,87), (1,41,68)(2,42,69)(3,22,70)(4,23,71)(5,24,72)(6,25,73)(7,26,74)(8,27,75)(9,28,76)(10,29,77)(11,30,78)(12,31,79)(13,32,80)(14,33,81)(15,34,82)(16,35,83)(17,36,84)(18,37,64)(19,38,65)(20,39,66)(21,40,67)(43,102,108)(44,103,109)(45,104,110)(46,105,111)(47,85,112)(48,86,113)(49,87,114)(50,88,115)(51,89,116)(52,90,117)(53,91,118)(54,92,119)(55,93,120)(56,94,121)(57,95,122)(58,96,123)(59,97,124)(60,98,125)(61,99,126)(62,100,106)(63,101,107), (1,60)(2,61)(3,62)(4,63)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,103)(74,104)(75,105)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93) );
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)], [(1,75,34),(2,76,35),(3,77,36),(4,78,37),(5,79,38),(6,80,39),(7,81,40),(8,82,41),(9,83,42),(10,84,22),(11,64,23),(12,65,24),(13,66,25),(14,67,26),(15,68,27),(16,69,28),(17,70,29),(18,71,30),(19,72,31),(20,73,32),(21,74,33),(43,122,88),(44,123,89),(45,124,90),(46,125,91),(47,126,92),(48,106,93),(49,107,94),(50,108,95),(51,109,96),(52,110,97),(53,111,98),(54,112,99),(55,113,100),(56,114,101),(57,115,102),(58,116,103),(59,117,104),(60,118,105),(61,119,85),(62,120,86),(63,121,87)], [(1,41,68),(2,42,69),(3,22,70),(4,23,71),(5,24,72),(6,25,73),(7,26,74),(8,27,75),(9,28,76),(10,29,77),(11,30,78),(12,31,79),(13,32,80),(14,33,81),(15,34,82),(16,35,83),(17,36,84),(18,37,64),(19,38,65),(20,39,66),(21,40,67),(43,102,108),(44,103,109),(45,104,110),(46,105,111),(47,85,112),(48,86,113),(49,87,114),(50,88,115),(51,89,116),(52,90,117),(53,91,118),(54,92,119),(55,93,120),(56,94,121),(57,95,122),(58,96,123),(59,97,124),(60,98,125),(61,99,126),(62,100,106),(63,101,107)], [(1,60),(2,61),(3,62),(4,63),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,121),(38,122),(39,123),(40,124),(41,125),(42,126),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,103),(74,104),(75,105),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91),(83,92),(84,93)]])
126 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 6A | 6B | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21L | 21M | ··· | 21CF | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 9 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 1 | ··· | 1 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
126 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 | C3⋊S3×C21 | C32×C21 | C7×C3⋊S3 | C3×C21 | C3×C3⋊S3 | C33 | C3⋊S3 | C32 | C3×C21 | C21 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 4 | 8 | 24 | 48 |
Matrix representation of C3⋊S3×C21 ►in GL4(𝔽43) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 6 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(43))| [23,0,0,0,0,23,0,0,0,0,13,0,0,0,0,13],[6,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,6],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3⋊S3×C21 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times C_{21} % in TeX
G:=Group("C3:S3xC21"); // GroupNames label
G:=SmallGroup(378,56);
// by ID
G=gap.SmallGroup(378,56);
# by ID
G:=PCGroup([5,-2,-3,-7,-3,-3,1683,6304]);
// Polycyclic
G:=Group<a,b,c,d|a^21=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations