direct product, abelian, monomial, 3-elementary
Aliases: C32×C12, SmallGroup(108,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C12 |
| C1 — C32×C12 |
| C1 — C32×C12 |
Generators and relations for C32×C12
G = < a,b,c | a3=b3=c12=1, ab=ba, ac=ca, bc=cb >
Subgroups: 84, all normal (6 characteristic)
C1, C2, C3, C4, C6, C32, C12, C3×C6, C33, C3×C12, C32×C6, C32×C12
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, C33, C3×C12, C32×C6, C32×C12
►Regular action on 108 points
Generators in S108
(1 81 108)(2 82 97)(3 83 98)(4 84 99)(5 73 100)(6 74 101)(7 75 102)(8 76 103)(9 77 104)(10 78 105)(11 79 106)(12 80 107)(13 66 25)(14 67 26)(15 68 27)(16 69 28)(17 70 29)(18 71 30)(19 72 31)(20 61 32)(21 62 33)(22 63 34)(23 64 35)(24 65 36)(37 85 59)(38 86 60)(39 87 49)(40 88 50)(41 89 51)(42 90 52)(43 91 53)(44 92 54)(45 93 55)(46 94 56)(47 95 57)(48 96 58)
(1 55 14)(2 56 15)(3 57 16)(4 58 17)(5 59 18)(6 60 19)(7 49 20)(8 50 21)(9 51 22)(10 52 23)(11 53 24)(12 54 13)(25 107 92)(26 108 93)(27 97 94)(28 98 95)(29 99 96)(30 100 85)(31 101 86)(32 102 87)(33 103 88)(34 104 89)(35 105 90)(36 106 91)(37 71 73)(38 72 74)(39 61 75)(40 62 76)(41 63 77)(42 64 78)(43 65 79)(44 66 80)(45 67 81)(46 68 82)(47 69 83)(48 70 84)
(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)
G:=sub<Sym(108)| (1,81,108)(2,82,97)(3,83,98)(4,84,99)(5,73,100)(6,74,101)(7,75,102)(8,76,103)(9,77,104)(10,78,105)(11,79,106)(12,80,107)(13,66,25)(14,67,26)(15,68,27)(16,69,28)(17,70,29)(18,71,30)(19,72,31)(20,61,32)(21,62,33)(22,63,34)(23,64,35)(24,65,36)(37,85,59)(38,86,60)(39,87,49)(40,88,50)(41,89,51)(42,90,52)(43,91,53)(44,92,54)(45,93,55)(46,94,56)(47,95,57)(48,96,58), (1,55,14)(2,56,15)(3,57,16)(4,58,17)(5,59,18)(6,60,19)(7,49,20)(8,50,21)(9,51,22)(10,52,23)(11,53,24)(12,54,13)(25,107,92)(26,108,93)(27,97,94)(28,98,95)(29,99,96)(30,100,85)(31,101,86)(32,102,87)(33,103,88)(34,104,89)(35,105,90)(36,106,91)(37,71,73)(38,72,74)(39,61,75)(40,62,76)(41,63,77)(42,64,78)(43,65,79)(44,66,80)(45,67,81)(46,68,82)(47,69,83)(48,70,84), (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)>;
G:=Group( (1,81,108)(2,82,97)(3,83,98)(4,84,99)(5,73,100)(6,74,101)(7,75,102)(8,76,103)(9,77,104)(10,78,105)(11,79,106)(12,80,107)(13,66,25)(14,67,26)(15,68,27)(16,69,28)(17,70,29)(18,71,30)(19,72,31)(20,61,32)(21,62,33)(22,63,34)(23,64,35)(24,65,36)(37,85,59)(38,86,60)(39,87,49)(40,88,50)(41,89,51)(42,90,52)(43,91,53)(44,92,54)(45,93,55)(46,94,56)(47,95,57)(48,96,58), (1,55,14)(2,56,15)(3,57,16)(4,58,17)(5,59,18)(6,60,19)(7,49,20)(8,50,21)(9,51,22)(10,52,23)(11,53,24)(12,54,13)(25,107,92)(26,108,93)(27,97,94)(28,98,95)(29,99,96)(30,100,85)(31,101,86)(32,102,87)(33,103,88)(34,104,89)(35,105,90)(36,106,91)(37,71,73)(38,72,74)(39,61,75)(40,62,76)(41,63,77)(42,64,78)(43,65,79)(44,66,80)(45,67,81)(46,68,82)(47,69,83)(48,70,84), (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) );
G=PermutationGroup([[(1,81,108),(2,82,97),(3,83,98),(4,84,99),(5,73,100),(6,74,101),(7,75,102),(8,76,103),(9,77,104),(10,78,105),(11,79,106),(12,80,107),(13,66,25),(14,67,26),(15,68,27),(16,69,28),(17,70,29),(18,71,30),(19,72,31),(20,61,32),(21,62,33),(22,63,34),(23,64,35),(24,65,36),(37,85,59),(38,86,60),(39,87,49),(40,88,50),(41,89,51),(42,90,52),(43,91,53),(44,92,54),(45,93,55),(46,94,56),(47,95,57),(48,96,58)], [(1,55,14),(2,56,15),(3,57,16),(4,58,17),(5,59,18),(6,60,19),(7,49,20),(8,50,21),(9,51,22),(10,52,23),(11,53,24),(12,54,13),(25,107,92),(26,108,93),(27,97,94),(28,98,95),(29,99,96),(30,100,85),(31,101,86),(32,102,87),(33,103,88),(34,104,89),(35,105,90),(36,106,91),(37,71,73),(38,72,74),(39,61,75),(40,62,76),(41,63,77),(42,64,78),(43,65,79),(44,66,80),(45,67,81),(46,68,82),(47,69,83),(48,70,84)], [(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)]])
C32×C12 is a maximal subgroup of
C33⋊7C8 C33⋊8Q8 C33⋊12D4
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 4A | 4B | 6A | ··· | 6Z | 12A | ··· | 12AZ |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C4 | C6 | C12 |
| kernel | C32×C12 | C32×C6 | C3×C12 | C33 | C3×C6 | C32 |
| # reps | 1 | 1 | 26 | 2 | 26 | 52 |
Matrix representation of C32×C12 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 1 |
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 1 |
| 11 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
G:=sub<GL(3,GF(13))| [1,0,0,0,9,0,0,0,1],[9,0,0,0,9,0,0,0,1],[11,0,0,0,2,0,0,0,2] >;
C32×C12 in GAP, Magma, Sage, TeX
C_3^2\times C_{12} % in TeX
G:=Group("C3^2xC12"); // GroupNames label
G:=SmallGroup(108,35);
// by ID
G=gap.SmallGroup(108,35);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-2,270]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations