direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C12×He3, C33⋊8C12, C12.1C33, C2.(C6×He3), (C3×C12)⋊C32, (C32×C12)⋊2C3, C32⋊3(C3×C12), (C6×He3).6C2, C6.2(C32×C6), C6.12(C2×He3), (C2×He3).14C6, (C32×C6).12C6, C3.1(C32×C12), (C3×C6).10(C3×C6), SmallGroup(324,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×He3
G = < a,b,c,d | a12=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Subgroups: 312 in 168 conjugacy classes, 96 normal (12 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C32, C32, C32, C12, C12, C12, C3×C6, C3×C6, C3×C6, He3, C33, C3×C12, C3×C12, C3×C12, C2×He3, C32×C6, C3×He3, C4×He3, C32×C12, C6×He3, C12×He3
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, He3, C33, C3×C12, C2×He3, C32×C6, C3×He3, C4×He3, C32×C12, C6×He3, C12×He3
►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 29 55)(2 30 56)(3 31 57)(4 32 58)(5 33 59)(6 34 60)(7 35 49)(8 36 50)(9 25 51)(10 26 52)(11 27 53)(12 28 54)(13 83 37)(14 84 38)(15 73 39)(16 74 40)(17 75 41)(18 76 42)(19 77 43)(20 78 44)(21 79 45)(22 80 46)(23 81 47)(24 82 48)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(85 89 93)(86 90 94)(87 91 95)(88 92 96)(97 101 105)(98 102 106)(99 103 107)(100 104 108)
(1 25 59)(2 26 60)(3 27 49)(4 28 50)(5 29 51)(6 30 52)(7 31 53)(8 32 54)(9 33 55)(10 34 56)(11 35 57)(12 36 58)(13 41 79)(14 42 80)(15 43 81)(16 44 82)(17 45 83)(18 46 84)(19 47 73)(20 48 74)(21 37 75)(22 38 76)(23 39 77)(24 40 78)(61 103 88)(62 104 89)(63 105 90)(64 106 91)(65 107 92)(66 108 93)(67 97 94)(68 98 95)(69 99 96)(70 100 85)(71 101 86)(72 102 87)
(1 86 84)(2 87 73)(3 88 74)(4 89 75)(5 90 76)(6 91 77)(7 92 78)(8 93 79)(9 94 80)(10 95 81)(11 96 82)(12 85 83)(13 32 66)(14 33 67)(15 34 68)(16 35 69)(17 36 70)(18 25 71)(19 26 72)(20 27 61)(21 28 62)(22 29 63)(23 30 64)(24 31 65)(37 50 104)(38 51 105)(39 52 106)(40 53 107)(41 54 108)(42 55 97)(43 56 98)(44 57 99)(45 58 100)(46 59 101)(47 60 102)(48 49 103)
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), (1,29,55)(2,30,56)(3,31,57)(4,32,58)(5,33,59)(6,34,60)(7,35,49)(8,36,50)(9,25,51)(10,26,52)(11,27,53)(12,28,54)(13,83,37)(14,84,38)(15,73,39)(16,74,40)(17,75,41)(18,76,42)(19,77,43)(20,78,44)(21,79,45)(22,80,46)(23,81,47)(24,82,48)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(85,89,93)(86,90,94)(87,91,95)(88,92,96)(97,101,105)(98,102,106)(99,103,107)(100,104,108), (1,25,59)(2,26,60)(3,27,49)(4,28,50)(5,29,51)(6,30,52)(7,31,53)(8,32,54)(9,33,55)(10,34,56)(11,35,57)(12,36,58)(13,41,79)(14,42,80)(15,43,81)(16,44,82)(17,45,83)(18,46,84)(19,47,73)(20,48,74)(21,37,75)(22,38,76)(23,39,77)(24,40,78)(61,103,88)(62,104,89)(63,105,90)(64,106,91)(65,107,92)(66,108,93)(67,97,94)(68,98,95)(69,99,96)(70,100,85)(71,101,86)(72,102,87), (1,86,84)(2,87,73)(3,88,74)(4,89,75)(5,90,76)(6,91,77)(7,92,78)(8,93,79)(9,94,80)(10,95,81)(11,96,82)(12,85,83)(13,32,66)(14,33,67)(15,34,68)(16,35,69)(17,36,70)(18,25,71)(19,26,72)(20,27,61)(21,28,62)(22,29,63)(23,30,64)(24,31,65)(37,50,104)(38,51,105)(39,52,106)(40,53,107)(41,54,108)(42,55,97)(43,56,98)(44,57,99)(45,58,100)(46,59,101)(47,60,102)(48,49,103)>;
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), (1,29,55)(2,30,56)(3,31,57)(4,32,58)(5,33,59)(6,34,60)(7,35,49)(8,36,50)(9,25,51)(10,26,52)(11,27,53)(12,28,54)(13,83,37)(14,84,38)(15,73,39)(16,74,40)(17,75,41)(18,76,42)(19,77,43)(20,78,44)(21,79,45)(22,80,46)(23,81,47)(24,82,48)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(85,89,93)(86,90,94)(87,91,95)(88,92,96)(97,101,105)(98,102,106)(99,103,107)(100,104,108), (1,25,59)(2,26,60)(3,27,49)(4,28,50)(5,29,51)(6,30,52)(7,31,53)(8,32,54)(9,33,55)(10,34,56)(11,35,57)(12,36,58)(13,41,79)(14,42,80)(15,43,81)(16,44,82)(17,45,83)(18,46,84)(19,47,73)(20,48,74)(21,37,75)(22,38,76)(23,39,77)(24,40,78)(61,103,88)(62,104,89)(63,105,90)(64,106,91)(65,107,92)(66,108,93)(67,97,94)(68,98,95)(69,99,96)(70,100,85)(71,101,86)(72,102,87), (1,86,84)(2,87,73)(3,88,74)(4,89,75)(5,90,76)(6,91,77)(7,92,78)(8,93,79)(9,94,80)(10,95,81)(11,96,82)(12,85,83)(13,32,66)(14,33,67)(15,34,68)(16,35,69)(17,36,70)(18,25,71)(19,26,72)(20,27,61)(21,28,62)(22,29,63)(23,30,64)(24,31,65)(37,50,104)(38,51,105)(39,52,106)(40,53,107)(41,54,108)(42,55,97)(43,56,98)(44,57,99)(45,58,100)(46,59,101)(47,60,102)(48,49,103) );
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)], [(1,29,55),(2,30,56),(3,31,57),(4,32,58),(5,33,59),(6,34,60),(7,35,49),(8,36,50),(9,25,51),(10,26,52),(11,27,53),(12,28,54),(13,83,37),(14,84,38),(15,73,39),(16,74,40),(17,75,41),(18,76,42),(19,77,43),(20,78,44),(21,79,45),(22,80,46),(23,81,47),(24,82,48),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(85,89,93),(86,90,94),(87,91,95),(88,92,96),(97,101,105),(98,102,106),(99,103,107),(100,104,108)], [(1,25,59),(2,26,60),(3,27,49),(4,28,50),(5,29,51),(6,30,52),(7,31,53),(8,32,54),(9,33,55),(10,34,56),(11,35,57),(12,36,58),(13,41,79),(14,42,80),(15,43,81),(16,44,82),(17,45,83),(18,46,84),(19,47,73),(20,48,74),(21,37,75),(22,38,76),(23,39,77),(24,40,78),(61,103,88),(62,104,89),(63,105,90),(64,106,91),(65,107,92),(66,108,93),(67,97,94),(68,98,95),(69,99,96),(70,100,85),(71,101,86),(72,102,87)], [(1,86,84),(2,87,73),(3,88,74),(4,89,75),(5,90,76),(6,91,77),(7,92,78),(8,93,79),(9,94,80),(10,95,81),(11,96,82),(12,85,83),(13,32,66),(14,33,67),(15,34,68),(16,35,69),(17,36,70),(18,25,71),(19,26,72),(20,27,61),(21,28,62),(22,29,63),(23,30,64),(24,31,65),(37,50,104),(38,51,105),(39,52,106),(40,53,107),(41,54,108),(42,55,97),(43,56,98),(44,57,99),(45,58,100),(46,59,101),(47,60,102),(48,49,103)]])
132 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3AF | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6AF | 12A | ··· | 12P | 12Q | ··· | 12BL |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | He3 | C2×He3 | C4×He3 |
| kernel | C12×He3 | C6×He3 | C4×He3 | C32×C12 | C3×He3 | C2×He3 | C32×C6 | He3 | C33 | C12 | C6 | C3 |
| # reps | 1 | 1 | 18 | 8 | 2 | 18 | 8 | 36 | 16 | 6 | 6 | 12 |
Matrix representation of C12×He3 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 12 | 5 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 5 | 5 | 12 |
| 0 | 0 | 0 | 3 |
| 0 | 11 | 4 | 8 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[3,0,0,0,0,3,0,0,0,12,9,0,0,5,0,1],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,5,0,11,0,5,0,4,0,12,3,8] >;
C12×He3 in GAP, Magma, Sage, TeX
C_{12}\times {\rm He}_3 % in TeX
G:=Group("C12xHe3"); // GroupNames label
G:=SmallGroup(324,106);
// by ID
G=gap.SmallGroup(324,106);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,-3,324,1034]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations