direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C54, C6⋊C54, C3⋊(C2×C54), (S3×C9).C6, (S3×C6).C9, (C3×C54)⋊1C2, (C3×S3).C18, (S3×C18).C3, C9.4(S3×C6), C6.9(S3×C9), C3.4(S3×C18), (C3×C6).6C18, (C3×C27)⋊2C22, C18.11(C3×S3), (C3×C18).22C6, C32.2(C2×C18), (C3×C9).5(C2×C6), SmallGroup(324,66)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C54 |
Generators and relations for S3×C54
G = < a,b,c | a54=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C54
►On 108 points
Generators in S108
(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)
(1 19 37)(2 20 38)(3 21 39)(4 22 40)(5 23 41)(6 24 42)(7 25 43)(8 26 44)(9 27 45)(10 28 46)(11 29 47)(12 30 48)(13 31 49)(14 32 50)(15 33 51)(16 34 52)(17 35 53)(18 36 54)(55 91 73)(56 92 74)(57 93 75)(58 94 76)(59 95 77)(60 96 78)(61 97 79)(62 98 80)(63 99 81)(64 100 82)(65 101 83)(66 102 84)(67 103 85)(68 104 86)(69 105 87)(70 106 88)(71 107 89)(72 108 90)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)
G:=sub<Sym(108)| (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), (1,19,37)(2,20,38)(3,21,39)(4,22,40)(5,23,41)(6,24,42)(7,25,43)(8,26,44)(9,27,45)(10,28,46)(11,29,47)(12,30,48)(13,31,49)(14,32,50)(15,33,51)(16,34,52)(17,35,53)(18,36,54)(55,91,73)(56,92,74)(57,93,75)(58,94,76)(59,95,77)(60,96,78)(61,97,79)(62,98,80)(63,99,81)(64,100,82)(65,101,83)(66,102,84)(67,103,85)(68,104,86)(69,105,87)(70,106,88)(71,107,89)(72,108,90), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)>;
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), (1,19,37)(2,20,38)(3,21,39)(4,22,40)(5,23,41)(6,24,42)(7,25,43)(8,26,44)(9,27,45)(10,28,46)(11,29,47)(12,30,48)(13,31,49)(14,32,50)(15,33,51)(16,34,52)(17,35,53)(18,36,54)(55,91,73)(56,92,74)(57,93,75)(58,94,76)(59,95,77)(60,96,78)(61,97,79)(62,98,80)(63,99,81)(64,100,82)(65,101,83)(66,102,84)(67,103,85)(68,104,86)(69,105,87)(70,106,88)(71,107,89)(72,108,90), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73) );
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)], [(1,19,37),(2,20,38),(3,21,39),(4,22,40),(5,23,41),(6,24,42),(7,25,43),(8,26,44),(9,27,45),(10,28,46),(11,29,47),(12,30,48),(13,31,49),(14,32,50),(15,33,51),(16,34,52),(17,35,53),(18,36,54),(55,91,73),(56,92,74),(57,93,75),(58,94,76),(59,95,77),(60,96,78),(61,97,79),(62,98,80),(63,99,81),(64,100,82),(65,101,83),(66,102,84),(67,103,85),(68,104,86),(69,105,87),(70,106,88),(71,107,89),(72,108,90)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,85),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,97),(25,98),(26,99),(27,100),(28,101),(29,102),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 9A | ··· | 9F | 9G | ··· | 9L | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18X | 27A | ··· | 27R | 27S | ··· | 27AJ | 54A | ··· | 54R | 54S | ··· | 54AJ | 54AK | ··· | 54BT |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 | 54 | ··· | 54 | 54 | ··· | 54 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | C27 | C54 | C54 | S3 | D6 | C3×S3 | S3×C6 | S3×C9 | S3×C18 | S3×C27 | S3×C54 |
| kernel | S3×C54 | S3×C27 | C3×C54 | S3×C18 | S3×C9 | C3×C18 | S3×C6 | C3×S3 | C3×C6 | D6 | S3 | C6 | C54 | C27 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 6 | 12 | 6 | 18 | 36 | 18 | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 |
Matrix representation of S3×C54 ►in GL3(𝔽109) generated by
| 106 | 0 | 0 |
| 0 | 21 | 0 |
| 0 | 0 | 21 |
| 1 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 45 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(109))| [106,0,0,0,21,0,0,0,21],[1,0,0,0,63,0,0,0,45],[1,0,0,0,0,1,0,1,0] >;
S3×C54 in GAP, Magma, Sage, TeX
S_3\times C_{54} % in TeX
G:=Group("S3xC54"); // GroupNames label
G:=SmallGroup(324,66);
// by ID
G=gap.SmallGroup(324,66);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,93,7781]);
// Polycyclic
G:=Group<a,b,c|a^54=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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