direct product, abelian, monomial
Aliases: C3×C62, SmallGroup(108,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C62 |
| C1 — C3×C62 |
| C1 — C3×C62 |
Generators and relations for C3×C62
G = < a,b,c | a3=b6=c6=1, ab=ba, ac=ca, bc=cb >
Subgroups: 140, all normal (4 characteristic)
C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C33, C62, C32×C6, C3×C62
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C33, C62, C32×C6, C3×C62
►Regular action on 108 points
Generators in S108
(1 83 93)(2 84 94)(3 79 95)(4 80 96)(5 81 91)(6 82 92)(7 88 48)(8 89 43)(9 90 44)(10 85 45)(11 86 46)(12 87 47)(13 31 23)(14 32 24)(15 33 19)(16 34 20)(17 35 21)(18 36 22)(25 68 73)(26 69 74)(27 70 75)(28 71 76)(29 72 77)(30 67 78)(37 100 51)(38 101 52)(39 102 53)(40 97 54)(41 98 49)(42 99 50)(55 62 105)(56 63 106)(57 64 107)(58 65 108)(59 66 103)(60 61 104)
(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 75 38 45 18 103)(2 76 39 46 13 104)(3 77 40 47 14 105)(4 78 41 48 15 106)(5 73 42 43 16 107)(6 74 37 44 17 108)(7 33 56 80 30 98)(8 34 57 81 25 99)(9 35 58 82 26 100)(10 36 59 83 27 101)(11 31 60 84 28 102)(12 32 55 79 29 97)(19 63 96 67 49 88)(20 64 91 68 50 89)(21 65 92 69 51 90)(22 66 93 70 52 85)(23 61 94 71 53 86)(24 62 95 72 54 87)
G:=sub<Sym(108)| (1,83,93)(2,84,94)(3,79,95)(4,80,96)(5,81,91)(6,82,92)(7,88,48)(8,89,43)(9,90,44)(10,85,45)(11,86,46)(12,87,47)(13,31,23)(14,32,24)(15,33,19)(16,34,20)(17,35,21)(18,36,22)(25,68,73)(26,69,74)(27,70,75)(28,71,76)(29,72,77)(30,67,78)(37,100,51)(38,101,52)(39,102,53)(40,97,54)(41,98,49)(42,99,50)(55,62,105)(56,63,106)(57,64,107)(58,65,108)(59,66,103)(60,61,104), (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), (1,75,38,45,18,103)(2,76,39,46,13,104)(3,77,40,47,14,105)(4,78,41,48,15,106)(5,73,42,43,16,107)(6,74,37,44,17,108)(7,33,56,80,30,98)(8,34,57,81,25,99)(9,35,58,82,26,100)(10,36,59,83,27,101)(11,31,60,84,28,102)(12,32,55,79,29,97)(19,63,96,67,49,88)(20,64,91,68,50,89)(21,65,92,69,51,90)(22,66,93,70,52,85)(23,61,94,71,53,86)(24,62,95,72,54,87)>;
G:=Group( (1,83,93)(2,84,94)(3,79,95)(4,80,96)(5,81,91)(6,82,92)(7,88,48)(8,89,43)(9,90,44)(10,85,45)(11,86,46)(12,87,47)(13,31,23)(14,32,24)(15,33,19)(16,34,20)(17,35,21)(18,36,22)(25,68,73)(26,69,74)(27,70,75)(28,71,76)(29,72,77)(30,67,78)(37,100,51)(38,101,52)(39,102,53)(40,97,54)(41,98,49)(42,99,50)(55,62,105)(56,63,106)(57,64,107)(58,65,108)(59,66,103)(60,61,104), (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), (1,75,38,45,18,103)(2,76,39,46,13,104)(3,77,40,47,14,105)(4,78,41,48,15,106)(5,73,42,43,16,107)(6,74,37,44,17,108)(7,33,56,80,30,98)(8,34,57,81,25,99)(9,35,58,82,26,100)(10,36,59,83,27,101)(11,31,60,84,28,102)(12,32,55,79,29,97)(19,63,96,67,49,88)(20,64,91,68,50,89)(21,65,92,69,51,90)(22,66,93,70,52,85)(23,61,94,71,53,86)(24,62,95,72,54,87) );
G=PermutationGroup([[(1,83,93),(2,84,94),(3,79,95),(4,80,96),(5,81,91),(6,82,92),(7,88,48),(8,89,43),(9,90,44),(10,85,45),(11,86,46),(12,87,47),(13,31,23),(14,32,24),(15,33,19),(16,34,20),(17,35,21),(18,36,22),(25,68,73),(26,69,74),(27,70,75),(28,71,76),(29,72,77),(30,67,78),(37,100,51),(38,101,52),(39,102,53),(40,97,54),(41,98,49),(42,99,50),(55,62,105),(56,63,106),(57,64,107),(58,65,108),(59,66,103),(60,61,104)], [(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)], [(1,75,38,45,18,103),(2,76,39,46,13,104),(3,77,40,47,14,105),(4,78,41,48,15,106),(5,73,42,43,16,107),(6,74,37,44,17,108),(7,33,56,80,30,98),(8,34,57,81,25,99),(9,35,58,82,26,100),(10,36,59,83,27,101),(11,31,60,84,28,102),(12,32,55,79,29,97),(19,63,96,67,49,88),(20,64,91,68,50,89),(21,65,92,69,51,90),(22,66,93,70,52,85),(23,61,94,71,53,86),(24,62,95,72,54,87)]])
C3×C62 is a maximal subgroup of
C33⋊15D4 C62⋊C9 C33⋊2A4
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3Z | 6A | ··· | 6BZ |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C3 | C6 |
| kernel | C3×C62 | C32×C6 | C62 | C3×C6 |
| # reps | 1 | 3 | 26 | 78 |
Matrix representation of C3×C62 ►in GL3(𝔽7) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 6 |
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 5 |
G:=sub<GL(3,GF(7))| [4,0,0,0,4,0,0,0,4],[1,0,0,0,2,0,0,0,6],[1,0,0,0,3,0,0,0,5] >;
C3×C62 in GAP, Magma, Sage, TeX
C_3\times C_6^2
% in TeX
G:=Group("C3xC6^2"); // GroupNames label
G:=SmallGroup(108,45);
// by ID
G=gap.SmallGroup(108,45);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3]);
// Polycyclic
G:=Group<a,b,c|a^3=b^6=c^6=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations