direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.3Q8, C12.18Dic6, C62.181C23, C6.5(C6×Q8), C12.3(C3×Q8), C4⋊Dic3.7C6, (C3×C12).15Q8, C4.3(C3×Dic6), C2.8(C6×Dic6), (C2×C12).235D6, Dic3⋊C4.3C6, (C4×Dic3).2C6, C6.52(C2×Dic6), (C6×C12).192C22, C6.56(Q8⋊3S3), (Dic3×C12).11C2, C6.118(D4⋊2S3), C32⋊12(C42.C2), (C6×Dic3).124C22, (C3×C4⋊C4).7C6, C4⋊C4.6(C3×S3), (C3×C4⋊C4).29S3, (C2×C4).42(S3×C6), C6.25(C3×C4○D4), C22.48(S3×C2×C6), (C3×C6).48(C2×Q8), C3⋊3(C3×C42.C2), (C2×C12).22(C2×C6), C2.4(C3×Q8⋊3S3), C2.12(C3×D4⋊2S3), (C32×C4⋊C4).10C2, (C3×C4⋊Dic3).26C2, (C2×C6).36(C22×C6), (C3×C6).132(C4○D4), (C3×Dic3⋊C4).11C2, (C2×C6).314(C22×S3), (C2×Dic3).27(C2×C6), SmallGroup(288,661)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.3Q8
G = < a,b,c,d | a3=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b6c-1 >
Subgroups: 242 in 125 conjugacy classes, 66 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×6], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C32, Dic3 [×4], C12 [×4], C12 [×12], C2×C6 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×5], C3×C6 [×3], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×7], C42.C2, C3×Dic3 [×4], C3×C12 [×2], C3×C12 [×2], C62, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×6], C6×Dic3 [×2], C6×Dic3 [×2], C6×C12, C6×C12 [×2], C12.3Q8, C3×C42.C2, Dic3×C12, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×C4⋊Dic3 [×2], C32×C4⋊C4, C3×C12.3Q8
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×Q8, C4○D4 [×2], C3×S3, Dic6 [×2], C3×Q8 [×2], C22×S3, C22×C6, C42.C2, S3×C6 [×3], C2×Dic6, D4⋊2S3, Q8⋊3S3, C6×Q8, C3×C4○D4 [×2], C3×Dic6 [×2], S3×C2×C6, C12.3Q8, C3×C42.C2, C6×Dic6, C3×D4⋊2S3, C3×Q8⋊3S3, C3×C12.3Q8
►On 96 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(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 39 23 60)(2 46 24 55)(3 41 13 50)(4 48 14 57)(5 43 15 52)(6 38 16 59)(7 45 17 54)(8 40 18 49)(9 47 19 56)(10 42 20 51)(11 37 21 58)(12 44 22 53)(25 77 70 86)(26 84 71 93)(27 79 72 88)(28 74 61 95)(29 81 62 90)(30 76 63 85)(31 83 64 92)(32 78 65 87)(33 73 66 94)(34 80 67 89)(35 75 68 96)(36 82 69 91)
(1 74 23 95)(2 79 24 88)(3 84 13 93)(4 77 14 86)(5 82 15 91)(6 75 16 96)(7 80 17 89)(8 73 18 94)(9 78 19 87)(10 83 20 92)(11 76 21 85)(12 81 22 90)(25 51 70 42)(26 56 71 47)(27 49 72 40)(28 54 61 45)(29 59 62 38)(30 52 63 43)(31 57 64 48)(32 50 65 41)(33 55 66 46)(34 60 67 39)(35 53 68 44)(36 58 69 37)
G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,39,23,60)(2,46,24,55)(3,41,13,50)(4,48,14,57)(5,43,15,52)(6,38,16,59)(7,45,17,54)(8,40,18,49)(9,47,19,56)(10,42,20,51)(11,37,21,58)(12,44,22,53)(25,77,70,86)(26,84,71,93)(27,79,72,88)(28,74,61,95)(29,81,62,90)(30,76,63,85)(31,83,64,92)(32,78,65,87)(33,73,66,94)(34,80,67,89)(35,75,68,96)(36,82,69,91), (1,74,23,95)(2,79,24,88)(3,84,13,93)(4,77,14,86)(5,82,15,91)(6,75,16,96)(7,80,17,89)(8,73,18,94)(9,78,19,87)(10,83,20,92)(11,76,21,85)(12,81,22,90)(25,51,70,42)(26,56,71,47)(27,49,72,40)(28,54,61,45)(29,59,62,38)(30,52,63,43)(31,57,64,48)(32,50,65,41)(33,55,66,46)(34,60,67,39)(35,53,68,44)(36,58,69,37)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,39,23,60)(2,46,24,55)(3,41,13,50)(4,48,14,57)(5,43,15,52)(6,38,16,59)(7,45,17,54)(8,40,18,49)(9,47,19,56)(10,42,20,51)(11,37,21,58)(12,44,22,53)(25,77,70,86)(26,84,71,93)(27,79,72,88)(28,74,61,95)(29,81,62,90)(30,76,63,85)(31,83,64,92)(32,78,65,87)(33,73,66,94)(34,80,67,89)(35,75,68,96)(36,82,69,91), (1,74,23,95)(2,79,24,88)(3,84,13,93)(4,77,14,86)(5,82,15,91)(6,75,16,96)(7,80,17,89)(8,73,18,94)(9,78,19,87)(10,83,20,92)(11,76,21,85)(12,81,22,90)(25,51,70,42)(26,56,71,47)(27,49,72,40)(28,54,61,45)(29,59,62,38)(30,52,63,43)(31,57,64,48)(32,50,65,41)(33,55,66,46)(34,60,67,39)(35,53,68,44)(36,58,69,37) );
G=PermutationGroup([(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(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)], [(1,39,23,60),(2,46,24,55),(3,41,13,50),(4,48,14,57),(5,43,15,52),(6,38,16,59),(7,45,17,54),(8,40,18,49),(9,47,19,56),(10,42,20,51),(11,37,21,58),(12,44,22,53),(25,77,70,86),(26,84,71,93),(27,79,72,88),(28,74,61,95),(29,81,62,90),(30,76,63,85),(31,83,64,92),(32,78,65,87),(33,73,66,94),(34,80,67,89),(35,75,68,96),(36,82,69,91)], [(1,74,23,95),(2,79,24,88),(3,84,13,93),(4,77,14,86),(5,82,15,91),(6,75,16,96),(7,80,17,89),(8,73,18,94),(9,78,19,87),(10,83,20,92),(11,76,21,85),(12,81,22,90),(25,51,70,42),(26,56,71,47),(27,49,72,40),(28,54,61,45),(29,59,62,38),(30,52,63,43),(31,57,64,48),(32,50,65,41),(33,55,66,46),(34,60,67,39),(35,53,68,44),(36,58,69,37)])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | ··· | 6O | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | ··· | 12AH | 12AI | 12AJ | 12AK | 12AL |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | Q8 | D6 | C4○D4 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | C3×C4○D4 | C3×Dic6 | D4⋊2S3 | Q8⋊3S3 | C3×D4⋊2S3 | C3×Q8⋊3S3 |
| kernel | C3×C12.3Q8 | Dic3×C12 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C32×C4⋊C4 | C12.3Q8 | C4×Dic3 | Dic3⋊C4 | C4⋊Dic3 | C3×C4⋊C4 | C3×C4⋊C4 | C3×C12 | C2×C12 | C3×C6 | C4⋊C4 | C12 | C12 | C2×C4 | C6 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 2 | 2 | 4 | 6 | 2 | 1 | 2 | 3 | 4 | 2 | 4 | 4 | 6 | 8 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C12.3Q8 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 2 | 3 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[0,1,0,0,12,0,0,0,0,0,9,2,0,0,0,3],[0,1,0,0,1,0,0,0,0,0,8,1,0,0,0,5],[0,5,0,0,8,0,0,0,0,0,8,0,0,0,2,5] >;
C3×C12.3Q8 in GAP, Magma, Sage, TeX
C_3\times C_{12}._3Q_8 % in TeX
G:=Group("C3xC12.3Q8"); // GroupNames label
G:=SmallGroup(288,661);
// by ID
G=gap.SmallGroup(288,661);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,1598,555,142,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations