direct product, metabelian, supersoluble, monomial
Aliases: C2×C6×D12, C62⋊20D4, C62.267C23, C6⋊1(C6×D4), (C2×C12)⋊31D6, (S3×C23)⋊6C6, (S3×C6)⋊8C23, (C3×C12)⋊8C23, C12⋊9(C22×S3), C12⋊2(C22×C6), D6⋊1(C22×C6), C6.3(C23×C6), C32⋊9(C22×D4), (C6×C12)⋊34C22, (C22×C12)⋊11C6, (C22×C12)⋊16S3, C23.45(S3×C6), (C3×C6).40C24, C6.71(S3×C23), (C22×C6).176D6, (C2×C62).117C22, C4⋊2(S3×C2×C6), C3⋊1(D4×C2×C6), (C2×C6×C12)⋊13C2, (C2×C4)⋊9(S3×C6), (C3×C6)⋊8(C2×D4), (C2×C6)⋊9(C3×D4), (S3×C22×C6)⋊8C2, (C2×C12)⋊12(C2×C6), C2.4(S3×C22×C6), (S3×C2×C6)⋊19C22, (C22×C4)⋊9(C3×S3), C22.30(S3×C2×C6), (C22×S3)⋊6(C2×C6), (C2×C6).69(C22×C6), (C22×C6).70(C2×C6), (C2×C6).347(C22×S3), SmallGroup(288,990)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6×D12
G = < a,b,c,d | a2=b6=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1210 in 499 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C32, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×D4, C3×C12, S3×C6, S3×C6, C62, C2×D12, C22×C12, C22×C12, C6×D4, S3×C23, C23×C6, C3×D12, C6×C12, S3×C2×C6, S3×C2×C6, C2×C62, C22×D12, D4×C2×C6, C6×D12, C2×C6×C12, S3×C22×C6, C2×C6×D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C24, C3×S3, D12, C3×D4, C22×S3, C22×C6, C22×D4, S3×C6, C2×D12, C6×D4, S3×C23, C23×C6, C3×D12, S3×C2×C6, C22×D12, D4×C2×C6, C6×D12, S3×C22×C6, C2×C6×D12
►On 96 points
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 95)(14 96)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 73)(46 74)(47 75)(48 76)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 61)(57 62)(58 63)(59 64)(60 65)
(1 24 5 16 9 20)(2 13 6 17 10 21)(3 14 7 18 11 22)(4 15 8 19 12 23)(25 88 29 92 33 96)(26 89 30 93 34 85)(27 90 31 94 35 86)(28 91 32 95 36 87)(37 60 45 56 41 52)(38 49 46 57 42 53)(39 50 47 58 43 54)(40 51 48 59 44 55)(61 81 69 77 65 73)(62 82 70 78 66 74)(63 83 71 79 67 75)(64 84 72 80 68 76)
(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)
(1 75)(2 74)(3 73)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 62)(14 61)(15 72)(16 71)(17 70)(18 69)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 41)(26 40)(27 39)(28 38)(29 37)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 96)(57 95)(58 94)(59 93)(60 92)
G:=sub<Sym(96)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,95)(14,96)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,73)(46,74)(47,75)(48,76)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,61)(57,62)(58,63)(59,64)(60,65), (1,24,5,16,9,20)(2,13,6,17,10,21)(3,14,7,18,11,22)(4,15,8,19,12,23)(25,88,29,92,33,96)(26,89,30,93,34,85)(27,90,31,94,35,86)(28,91,32,95,36,87)(37,60,45,56,41,52)(38,49,46,57,42,53)(39,50,47,58,43,54)(40,51,48,59,44,55)(61,81,69,77,65,73)(62,82,70,78,66,74)(63,83,71,79,67,75)(64,84,72,80,68,76), (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), (1,75)(2,74)(3,73)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,62)(14,61)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,41)(26,40)(27,39)(28,38)(29,37)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,96)(57,95)(58,94)(59,93)(60,92)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,95)(14,96)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,73)(46,74)(47,75)(48,76)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,61)(57,62)(58,63)(59,64)(60,65), (1,24,5,16,9,20)(2,13,6,17,10,21)(3,14,7,18,11,22)(4,15,8,19,12,23)(25,88,29,92,33,96)(26,89,30,93,34,85)(27,90,31,94,35,86)(28,91,32,95,36,87)(37,60,45,56,41,52)(38,49,46,57,42,53)(39,50,47,58,43,54)(40,51,48,59,44,55)(61,81,69,77,65,73)(62,82,70,78,66,74)(63,83,71,79,67,75)(64,84,72,80,68,76), (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), (1,75)(2,74)(3,73)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,62)(14,61)(15,72)(16,71)(17,70)(18,69)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,41)(26,40)(27,39)(28,38)(29,37)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,96)(57,95)(58,94)(59,93)(60,92) );
G=PermutationGroup([[(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,95),(14,96),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,92),(23,93),(24,94),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,73),(46,74),(47,75),(48,76),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,61),(57,62),(58,63),(59,64),(60,65)], [(1,24,5,16,9,20),(2,13,6,17,10,21),(3,14,7,18,11,22),(4,15,8,19,12,23),(25,88,29,92,33,96),(26,89,30,93,34,85),(27,90,31,94,35,86),(28,91,32,95,36,87),(37,60,45,56,41,52),(38,49,46,57,42,53),(39,50,47,58,43,54),(40,51,48,59,44,55),(61,81,69,77,65,73),(62,82,70,78,66,74),(63,83,71,79,67,75),(64,84,72,80,68,76)], [(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)], [(1,75),(2,74),(3,73),(4,84),(5,83),(6,82),(7,81),(8,80),(9,79),(10,78),(11,77),(12,76),(13,62),(14,61),(15,72),(16,71),(17,70),(18,69),(19,68),(20,67),(21,66),(22,65),(23,64),(24,63),(25,41),(26,40),(27,39),(28,38),(29,37),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,96),(57,95),(58,94),(59,93),(60,92)]])
108 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6N | 6O | ··· | 6AI | 6AJ | ··· | 6AY | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | ··· | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | C3×S3 | D12 | C3×D4 | S3×C6 | S3×C6 | C3×D12 |
| kernel | C2×C6×D12 | C6×D12 | C2×C6×C12 | S3×C22×C6 | C22×D12 | C2×D12 | C22×C12 | S3×C23 | C22×C12 | C62 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 2 | 2 | 24 | 2 | 4 | 1 | 4 | 6 | 1 | 2 | 8 | 8 | 12 | 2 | 16 |
Matrix representation of C2×C6×D12 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 10 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 8 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 |
| 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 0 | 9 | 4 |
| 0 | 0 | 0 | 6 | 4 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[10,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,9],[12,0,0,0,0,0,4,3,0,0,0,0,10,0,0,0,0,0,2,8,0,0,0,0,7],[1,0,0,0,0,0,3,3,0,0,0,6,10,0,0,0,0,0,9,6,0,0,0,4,4] >;
C2×C6×D12 in GAP, Magma, Sage, TeX
C_2\times C_6\times D_{12} % in TeX
G:=Group("C2xC6xD12"); // GroupNames label
G:=SmallGroup(288,990);
// by ID
G=gap.SmallGroup(288,990);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,1571,192,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^12=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