direct product, metabelian, supersoluble, monomial
Aliases: C3×C4.Dic6, 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×C4.Dic6
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, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C42.C2, C3×Dic3, C3×C12, C3×C12, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C6×Dic3, C6×Dic3, C6×C12, C6×C12, C4.Dic6, C3×C42.C2, Dic3×C12, C3×Dic3⋊C4, C3×C4⋊Dic3, C3×C4⋊Dic3, C32×C4⋊C4, C3×C4.Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C4○D4, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, C42.C2, S3×C6, C2×Dic6, D4⋊2S3, Q8⋊3S3, C6×Q8, C3×C4○D4, C3×Dic6, S3×C2×C6, C4.Dic6, C3×C42.C2, C6×Dic6, C3×D4⋊2S3, C3×Q8⋊3S3, C3×C4.Dic6
►On 96 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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 45 41)(38 46 42)(39 47 43)(40 48 44)(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 77 81)(74 78 82)(75 79 83)(76 80 84)(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 64 74 33)(2 71 75 28)(3 66 76 35)(4 61 77 30)(5 68 78 25)(6 63 79 32)(7 70 80 27)(8 65 81 34)(9 72 82 29)(10 67 83 36)(11 62 84 31)(12 69 73 26)(13 87 55 45)(14 94 56 40)(15 89 57 47)(16 96 58 42)(17 91 59 37)(18 86 60 44)(19 93 49 39)(20 88 50 46)(21 95 51 41)(22 90 52 48)(23 85 53 43)(24 92 54 38)
(1 21 74 51)(2 14 75 56)(3 19 76 49)(4 24 77 54)(5 17 78 59)(6 22 79 52)(7 15 80 57)(8 20 81 50)(9 13 82 55)(10 18 83 60)(11 23 84 53)(12 16 73 58)(25 85 68 43)(26 90 69 48)(27 95 70 41)(28 88 71 46)(29 93 72 39)(30 86 61 44)(31 91 62 37)(32 96 63 42)(33 89 64 47)(34 94 65 40)(35 87 66 45)(36 92 67 38)
G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(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), (1,64,74,33)(2,71,75,28)(3,66,76,35)(4,61,77,30)(5,68,78,25)(6,63,79,32)(7,70,80,27)(8,65,81,34)(9,72,82,29)(10,67,83,36)(11,62,84,31)(12,69,73,26)(13,87,55,45)(14,94,56,40)(15,89,57,47)(16,96,58,42)(17,91,59,37)(18,86,60,44)(19,93,49,39)(20,88,50,46)(21,95,51,41)(22,90,52,48)(23,85,53,43)(24,92,54,38), (1,21,74,51)(2,14,75,56)(3,19,76,49)(4,24,77,54)(5,17,78,59)(6,22,79,52)(7,15,80,57)(8,20,81,50)(9,13,82,55)(10,18,83,60)(11,23,84,53)(12,16,73,58)(25,85,68,43)(26,90,69,48)(27,95,70,41)(28,88,71,46)(29,93,72,39)(30,86,61,44)(31,91,62,37)(32,96,63,42)(33,89,64,47)(34,94,65,40)(35,87,66,45)(36,92,67,38)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(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), (1,64,74,33)(2,71,75,28)(3,66,76,35)(4,61,77,30)(5,68,78,25)(6,63,79,32)(7,70,80,27)(8,65,81,34)(9,72,82,29)(10,67,83,36)(11,62,84,31)(12,69,73,26)(13,87,55,45)(14,94,56,40)(15,89,57,47)(16,96,58,42)(17,91,59,37)(18,86,60,44)(19,93,49,39)(20,88,50,46)(21,95,51,41)(22,90,52,48)(23,85,53,43)(24,92,54,38), (1,21,74,51)(2,14,75,56)(3,19,76,49)(4,24,77,54)(5,17,78,59)(6,22,79,52)(7,15,80,57)(8,20,81,50)(9,13,82,55)(10,18,83,60)(11,23,84,53)(12,16,73,58)(25,85,68,43)(26,90,69,48)(27,95,70,41)(28,88,71,46)(29,93,72,39)(30,86,61,44)(31,91,62,37)(32,96,63,42)(33,89,64,47)(34,94,65,40)(35,87,66,45)(36,92,67,38) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(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)], [(1,64,74,33),(2,71,75,28),(3,66,76,35),(4,61,77,30),(5,68,78,25),(6,63,79,32),(7,70,80,27),(8,65,81,34),(9,72,82,29),(10,67,83,36),(11,62,84,31),(12,69,73,26),(13,87,55,45),(14,94,56,40),(15,89,57,47),(16,96,58,42),(17,91,59,37),(18,86,60,44),(19,93,49,39),(20,88,50,46),(21,95,51,41),(22,90,52,48),(23,85,53,43),(24,92,54,38)], [(1,21,74,51),(2,14,75,56),(3,19,76,49),(4,24,77,54),(5,17,78,59),(6,22,79,52),(7,15,80,57),(8,20,81,50),(9,13,82,55),(10,18,83,60),(11,23,84,53),(12,16,73,58),(25,85,68,43),(26,90,69,48),(27,95,70,41),(28,88,71,46),(29,93,72,39),(30,86,61,44),(31,91,62,37),(32,96,63,42),(33,89,64,47),(34,94,65,40),(35,87,66,45),(36,92,67,38)]])
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×C4.Dic6 | Dic3×C12 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C32×C4⋊C4 | C4.Dic6 | 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×C4.Dic6 ►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×C4.Dic6 in GAP, Magma, Sage, TeX
C_3\times C_4.{\rm Dic}_6 % in TeX
G:=Group("C3xC4.Dic6"); // 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