direct product, metabelian, supersoluble, monomial, A-group
Aliases: C32×D21, C33⋊1D7, C3⋊(C32×D7), C21⋊6(C3×S3), (C3×C21)⋊7S3, C21⋊5(C3×C6), C7⋊3(S3×C32), (C3×C21)⋊16C6, (C32×C21)⋊2C2, C32⋊3(C3×D7), SmallGroup(378,55)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C32×D21 |
Generators and relations for C32×D21
G = < a,b,c,d | a3=b3=c21=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 248 in 64 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C7, C32, C32, C32, D7, C3×S3, C3×C6, C21, C21, C21, C33, C3×D7, D21, S3×C32, C3×C21, C3×C21, C3×C21, C32×D7, C3×D21, C32×C21, C32×D21
Quotients: C1, C2, C3, S3, C6, C32, D7, C3×S3, C3×C6, C3×D7, D21, S3×C32, C32×D7, C3×D21, C32×D21
►On 126 points
Generators in S126
(1 28 52)(2 29 53)(3 30 54)(4 31 55)(5 32 56)(6 33 57)(7 34 58)(8 35 59)(9 36 60)(10 37 61)(11 38 62)(12 39 63)(13 40 43)(14 41 44)(15 42 45)(16 22 46)(17 23 47)(18 24 48)(19 25 49)(20 26 50)(21 27 51)(64 98 118)(65 99 119)(66 100 120)(67 101 121)(68 102 122)(69 103 123)(70 104 124)(71 105 125)(72 85 126)(73 86 106)(74 87 107)(75 88 108)(76 89 109)(77 90 110)(78 91 111)(79 92 112)(80 93 113)(81 94 114)(82 95 115)(83 96 116)(84 97 117)
(1 45 35)(2 46 36)(3 47 37)(4 48 38)(5 49 39)(6 50 40)(7 51 41)(8 52 42)(9 53 22)(10 54 23)(11 55 24)(12 56 25)(13 57 26)(14 58 27)(15 59 28)(16 60 29)(17 61 30)(18 62 31)(19 63 32)(20 43 33)(21 44 34)(64 125 91)(65 126 92)(66 106 93)(67 107 94)(68 108 95)(69 109 96)(70 110 97)(71 111 98)(72 112 99)(73 113 100)(74 114 101)(75 115 102)(76 116 103)(77 117 104)(78 118 105)(79 119 85)(80 120 86)(81 121 87)(82 122 88)(83 123 89)(84 124 90)
(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 93)(23 92)(24 91)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 126)(55 125)(56 124)(57 123)(58 122)(59 121)(60 120)(61 119)(62 118)(63 117)
G:=sub<Sym(126)| (1,28,52)(2,29,53)(3,30,54)(4,31,55)(5,32,56)(6,33,57)(7,34,58)(8,35,59)(9,36,60)(10,37,61)(11,38,62)(12,39,63)(13,40,43)(14,41,44)(15,42,45)(16,22,46)(17,23,47)(18,24,48)(19,25,49)(20,26,50)(21,27,51)(64,98,118)(65,99,119)(66,100,120)(67,101,121)(68,102,122)(69,103,123)(70,104,124)(71,105,125)(72,85,126)(73,86,106)(74,87,107)(75,88,108)(76,89,109)(77,90,110)(78,91,111)(79,92,112)(80,93,113)(81,94,114)(82,95,115)(83,96,116)(84,97,117), (1,45,35)(2,46,36)(3,47,37)(4,48,38)(5,49,39)(6,50,40)(7,51,41)(8,52,42)(9,53,22)(10,54,23)(11,55,24)(12,56,25)(13,57,26)(14,58,27)(15,59,28)(16,60,29)(17,61,30)(18,62,31)(19,63,32)(20,43,33)(21,44,34)(64,125,91)(65,126,92)(66,106,93)(67,107,94)(68,108,95)(69,109,96)(70,110,97)(71,111,98)(72,112,99)(73,113,100)(74,114,101)(75,115,102)(76,116,103)(77,117,104)(78,118,105)(79,119,85)(80,120,86)(81,121,87)(82,122,88)(83,123,89)(84,124,90), (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)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,105)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,126)(55,125)(56,124)(57,123)(58,122)(59,121)(60,120)(61,119)(62,118)(63,117)>;
G:=Group( (1,28,52)(2,29,53)(3,30,54)(4,31,55)(5,32,56)(6,33,57)(7,34,58)(8,35,59)(9,36,60)(10,37,61)(11,38,62)(12,39,63)(13,40,43)(14,41,44)(15,42,45)(16,22,46)(17,23,47)(18,24,48)(19,25,49)(20,26,50)(21,27,51)(64,98,118)(65,99,119)(66,100,120)(67,101,121)(68,102,122)(69,103,123)(70,104,124)(71,105,125)(72,85,126)(73,86,106)(74,87,107)(75,88,108)(76,89,109)(77,90,110)(78,91,111)(79,92,112)(80,93,113)(81,94,114)(82,95,115)(83,96,116)(84,97,117), (1,45,35)(2,46,36)(3,47,37)(4,48,38)(5,49,39)(6,50,40)(7,51,41)(8,52,42)(9,53,22)(10,54,23)(11,55,24)(12,56,25)(13,57,26)(14,58,27)(15,59,28)(16,60,29)(17,61,30)(18,62,31)(19,63,32)(20,43,33)(21,44,34)(64,125,91)(65,126,92)(66,106,93)(67,107,94)(68,108,95)(69,109,96)(70,110,97)(71,111,98)(72,112,99)(73,113,100)(74,114,101)(75,115,102)(76,116,103)(77,117,104)(78,118,105)(79,119,85)(80,120,86)(81,121,87)(82,122,88)(83,123,89)(84,124,90), (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)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,105)(32,104)(33,103)(34,102)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,126)(55,125)(56,124)(57,123)(58,122)(59,121)(60,120)(61,119)(62,118)(63,117) );
G=PermutationGroup([[(1,28,52),(2,29,53),(3,30,54),(4,31,55),(5,32,56),(6,33,57),(7,34,58),(8,35,59),(9,36,60),(10,37,61),(11,38,62),(12,39,63),(13,40,43),(14,41,44),(15,42,45),(16,22,46),(17,23,47),(18,24,48),(19,25,49),(20,26,50),(21,27,51),(64,98,118),(65,99,119),(66,100,120),(67,101,121),(68,102,122),(69,103,123),(70,104,124),(71,105,125),(72,85,126),(73,86,106),(74,87,107),(75,88,108),(76,89,109),(77,90,110),(78,91,111),(79,92,112),(80,93,113),(81,94,114),(82,95,115),(83,96,116),(84,97,117)], [(1,45,35),(2,46,36),(3,47,37),(4,48,38),(5,49,39),(6,50,40),(7,51,41),(8,52,42),(9,53,22),(10,54,23),(11,55,24),(12,56,25),(13,57,26),(14,58,27),(15,59,28),(16,60,29),(17,61,30),(18,62,31),(19,63,32),(20,43,33),(21,44,34),(64,125,91),(65,126,92),(66,106,93),(67,107,94),(68,108,95),(69,109,96),(70,110,97),(71,111,98),(72,112,99),(73,113,100),(74,114,101),(75,115,102),(76,116,103),(77,117,104),(78,118,105),(79,119,85),(80,120,86),(81,121,87),(82,122,88),(83,123,89),(84,124,90)], [(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),(22,93),(23,92),(24,91),(25,90),(26,89),(27,88),(28,87),(29,86),(30,85),(31,105),(32,104),(33,103),(34,102),(35,101),(36,100),(37,99),(38,98),(39,97),(40,96),(41,95),(42,94),(43,116),(44,115),(45,114),(46,113),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,106),(54,126),(55,125),(56,124),(57,123),(58,122),(59,121),(60,120),(61,119),(62,118),(63,117)]])
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 6A | ··· | 6H | 7A | 7B | 7C | 21A | ··· | 21BZ |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 7 | 7 | 7 | 21 | ··· | 21 |
| size | 1 | 21 | 1 | ··· | 1 | 2 | ··· | 2 | 21 | ··· | 21 | 2 | 2 | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | D7 | C3×S3 | C3×D7 | D21 | C3×D21 |
| kernel | C32×D21 | C32×C21 | C3×D21 | C3×C21 | C3×C21 | C33 | C21 | C32 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 1 | 3 | 8 | 24 | 6 | 48 |
Matrix representation of C32×D21 ►in GL3(𝔽43) generated by
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 1 | 0 | 0 |
| 0 | 31 | 9 |
| 0 | 0 | 25 |
| 1 | 0 | 0 |
| 0 | 5 | 33 |
| 0 | 11 | 38 |
G:=sub<GL(3,GF(43))| [1,0,0,0,6,0,0,0,6],[6,0,0,0,6,0,0,0,6],[1,0,0,0,31,0,0,9,25],[1,0,0,0,5,11,0,33,38] >;
C32×D21 in GAP, Magma, Sage, TeX
C_3^2\times D_{21} % in TeX
G:=Group("C3^2xD21"); // GroupNames label
G:=SmallGroup(378,55);
// by ID
G=gap.SmallGroup(378,55);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,723,8104]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^21=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