direct product, metabelian, supersoluble, monomial
Aliases: C3×C4.D12, C12.88D12, C62.188C23, (S3×C6)⋊9Q8, D6⋊2(C3×Q8), C6.8(C6×D4), D6⋊C4.3C6, C4⋊Dic3⋊6C6, C6.56(S3×Q8), C6.14(C6×Q8), (C2×Dic6)⋊7C6, C4.13(C3×D12), C2.10(C6×D12), C6.96(C2×D12), (C3×C12).81D4, C12.11(C3×D4), (C6×Dic6)⋊12C2, (C2×C12).236D6, C32⋊19(C22⋊Q8), (C6×C12).195C22, C6.120(D4⋊2S3), (C6×Dic3).130C22, (C3×C4⋊C4)⋊8C6, C4⋊C4⋊5(C3×S3), C2.7(C3×S3×Q8), (S3×C2×C4).2C6, (C3×C4⋊C4)⋊14S3, C3⋊3(C3×C22⋊Q8), (C32×C4⋊C4)⋊9C2, (S3×C2×C12).11C2, (C2×C4).44(S3×C6), (C3×D6⋊C4).9C2, C6.27(C3×C4○D4), C22.52(S3×C2×C6), (C3×C6).66(C2×Q8), (C2×C12).23(C2×C6), (C3×C4⋊Dic3)⋊30C2, (C3×C6).178(C2×D4), (S3×C2×C6).96C22, C2.13(C3×D4⋊2S3), (C2×C6).43(C22×C6), (C2×Dic3).8(C2×C6), (C3×C6).134(C4○D4), (C22×S3).23(C2×C6), (C2×C6).321(C22×S3), SmallGroup(288,668)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.D12
G = < a,b,c,d | a3=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 370 in 161 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×S3, C3×C6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C22⋊Q8, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×C12, C6×Q8, C3×Dic6, S3×C12, C6×Dic3, C6×Dic3, C6×C12, C6×C12, S3×C2×C6, C4.D12, C3×C22⋊Q8, C3×C4⋊Dic3, C3×D6⋊C4, C32×C4⋊C4, C6×Dic6, S3×C2×C12, C3×C4.D12
Quotients: C1, C2, C3, C22, S3, C6, D4, Q8, C23, D6, C2×C6, C2×D4, C2×Q8, C4○D4, C3×S3, D12, C3×D4, C3×Q8, C22×S3, C22×C6, C22⋊Q8, S3×C6, C2×D12, D4⋊2S3, S3×Q8, C6×D4, C6×Q8, C3×C4○D4, C3×D12, S3×C2×C6, C4.D12, C3×C22⋊Q8, C6×D12, C3×D4⋊2S3, C3×S3×Q8, C3×C4.D12
►On 96 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 75 60 15)(2 16 49 76)(3 77 50 17)(4 18 51 78)(5 79 52 19)(6 20 53 80)(7 81 54 21)(8 22 55 82)(9 83 56 23)(10 24 57 84)(11 73 58 13)(12 14 59 74)(25 48 96 72)(26 61 85 37)(27 38 86 62)(28 63 87 39)(29 40 88 64)(30 65 89 41)(31 42 90 66)(32 67 91 43)(33 44 92 68)(34 69 93 45)(35 46 94 70)(36 71 95 47)
(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 31 60 90)(2 89 49 30)(3 29 50 88)(4 87 51 28)(5 27 52 86)(6 85 53 26)(7 25 54 96)(8 95 55 36)(9 35 56 94)(10 93 57 34)(11 33 58 92)(12 91 59 32)(13 44 73 68)(14 67 74 43)(15 42 75 66)(16 65 76 41)(17 40 77 64)(18 63 78 39)(19 38 79 62)(20 61 80 37)(21 48 81 72)(22 71 82 47)(23 46 83 70)(24 69 84 45)
G:=sub<Sym(96)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,75,60,15)(2,16,49,76)(3,77,50,17)(4,18,51,78)(5,79,52,19)(6,20,53,80)(7,81,54,21)(8,22,55,82)(9,83,56,23)(10,24,57,84)(11,73,58,13)(12,14,59,74)(25,48,96,72)(26,61,85,37)(27,38,86,62)(28,63,87,39)(29,40,88,64)(30,65,89,41)(31,42,90,66)(32,67,91,43)(33,44,92,68)(34,69,93,45)(35,46,94,70)(36,71,95,47), (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), (1,31,60,90)(2,89,49,30)(3,29,50,88)(4,87,51,28)(5,27,52,86)(6,85,53,26)(7,25,54,96)(8,95,55,36)(9,35,56,94)(10,93,57,34)(11,33,58,92)(12,91,59,32)(13,44,73,68)(14,67,74,43)(15,42,75,66)(16,65,76,41)(17,40,77,64)(18,63,78,39)(19,38,79,62)(20,61,80,37)(21,48,81,72)(22,71,82,47)(23,46,83,70)(24,69,84,45)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,75,60,15)(2,16,49,76)(3,77,50,17)(4,18,51,78)(5,79,52,19)(6,20,53,80)(7,81,54,21)(8,22,55,82)(9,83,56,23)(10,24,57,84)(11,73,58,13)(12,14,59,74)(25,48,96,72)(26,61,85,37)(27,38,86,62)(28,63,87,39)(29,40,88,64)(30,65,89,41)(31,42,90,66)(32,67,91,43)(33,44,92,68)(34,69,93,45)(35,46,94,70)(36,71,95,47), (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), (1,31,60,90)(2,89,49,30)(3,29,50,88)(4,87,51,28)(5,27,52,86)(6,85,53,26)(7,25,54,96)(8,95,55,36)(9,35,56,94)(10,93,57,34)(11,33,58,92)(12,91,59,32)(13,44,73,68)(14,67,74,43)(15,42,75,66)(16,65,76,41)(17,40,77,64)(18,63,78,39)(19,38,79,62)(20,61,80,37)(21,48,81,72)(22,71,82,47)(23,46,83,70)(24,69,84,45) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(1,75,60,15),(2,16,49,76),(3,77,50,17),(4,18,51,78),(5,79,52,19),(6,20,53,80),(7,81,54,21),(8,22,55,82),(9,83,56,23),(10,24,57,84),(11,73,58,13),(12,14,59,74),(25,48,96,72),(26,61,85,37),(27,38,86,62),(28,63,87,39),(29,40,88,64),(30,65,89,41),(31,42,90,66),(32,67,91,43),(33,44,92,68),(34,69,93,45),(35,46,94,70),(36,71,95,47)], [(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)], [(1,31,60,90),(2,89,49,30),(3,29,50,88),(4,87,51,28),(5,27,52,86),(6,85,53,26),(7,25,54,96),(8,95,55,36),(9,35,56,94),(10,93,57,34),(11,33,58,92),(12,91,59,32),(13,44,73,68),(14,67,74,43),(15,42,75,66),(16,65,76,41),(17,40,77,64),(18,63,78,39),(19,38,79,62),(20,61,80,37),(21,48,81,72),(22,71,82,47),(23,46,83,70),(24,69,84,45)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | 12AB | 12AC | 12AD | 12AE | 12AF | 12AG | 12AH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | Q8 | D6 | C4○D4 | C3×S3 | D12 | C3×D4 | C3×Q8 | S3×C6 | C3×C4○D4 | C3×D12 | D4⋊2S3 | S3×Q8 | C3×D4⋊2S3 | C3×S3×Q8 |
| kernel | C3×C4.D12 | C3×C4⋊Dic3 | C3×D6⋊C4 | C32×C4⋊C4 | C6×Dic6 | S3×C2×C12 | C4.D12 | C4⋊Dic3 | D6⋊C4 | C3×C4⋊C4 | C2×Dic6 | S3×C2×C4 | C3×C4⋊C4 | C3×C12 | S3×C6 | C2×C12 | C3×C6 | C4⋊C4 | C12 | C12 | D6 | C2×C4 | C6 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 1 | 2 | 2 | 3 | 2 | 2 | 4 | 4 | 4 | 6 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C4.D12 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 11 | 9 |
| 6 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 11 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,4,11,0,0,2,9],[6,0,0,0,0,11,0,0,0,0,0,1,0,0,1,0],[0,6,0,0,11,0,0,0,0,0,3,7,0,0,6,10] >;
C3×C4.D12 in GAP, Magma, Sage, TeX
C_3\times C_4.D_{12} % in TeX
G:=Group("C3xC4.D12"); // GroupNames label
G:=SmallGroup(288,668);
// by ID
G=gap.SmallGroup(288,668);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,394,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^12=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations