direct product, metabelian, supersoluble, monomial
Aliases: C3×C12⋊D4, C12⋊9D12, C62.186C23, D6⋊C4⋊7C6, D6⋊2(C3×D4), (C3×C12)⋊9D4, C4⋊2(C3×D12), C12⋊1(C3×D4), C6.7(C6×D4), (S3×C6)⋊12D4, (C2×D12)⋊6C6, C2.9(C6×D12), (C6×D12)⋊27C2, C6.95(C2×D12), C6.185(S3×D4), (C2×C12).320D6, C32⋊18(C4⋊D4), (C6×C12).194C22, C6.60(Q8⋊3S3), (C6×Dic3).129C22, (S3×C2×C4)⋊1C6, (C3×C4⋊C4)⋊6C6, C4⋊C4⋊3(C3×S3), (S3×C2×C12)⋊5C2, (C3×C4⋊C4)⋊12S3, C2.13(C3×S3×D4), C3⋊2(C3×C4⋊D4), (C3×D6⋊C4)⋊19C2, (C32×C4⋊C4)⋊7C2, (C2×C12).7(C2×C6), (C2×C4).10(S3×C6), C6.34(C3×C4○D4), C22.50(S3×C2×C6), (C3×C6).177(C2×D4), (S3×C2×C6).57C22, C2.6(C3×Q8⋊3S3), (C22×S3).7(C2×C6), (C2×C6).41(C22×C6), (C3×C6).156(C4○D4), (C2×C6).319(C22×S3), (C2×Dic3).30(C2×C6), SmallGroup(288,666)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12⋊D4
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd=b-1, dcd=c-1 >
Subgroups: 562 in 201 conjugacy classes, 70 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C4⋊D4, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12, C22×C12, C6×D4, S3×C12, C3×D12, C6×Dic3, C6×C12, C6×C12, S3×C2×C6, S3×C2×C6, C12⋊D4, C3×C4⋊D4, C3×D6⋊C4, C32×C4⋊C4, S3×C2×C12, C6×D12, C6×D12, C3×C12⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C4⋊D4, S3×C6, C2×D12, S3×D4, Q8⋊3S3, C6×D4, C3×C4○D4, C3×D12, S3×C2×C6, C12⋊D4, C3×C4⋊D4, C6×D12, C3×S3×D4, C3×Q8⋊3S3, C3×C12⋊D4
►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 19 58 43)(2 14 59 38)(3 21 60 45)(4 16 49 40)(5 23 50 47)(6 18 51 42)(7 13 52 37)(8 20 53 44)(9 15 54 39)(10 22 55 46)(11 17 56 41)(12 24 57 48)(25 72 79 85)(26 67 80 92)(27 62 81 87)(28 69 82 94)(29 64 83 89)(30 71 84 96)(31 66 73 91)(32 61 74 86)(33 68 75 93)(34 63 76 88)(35 70 77 95)(36 65 78 90)
(1 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 72)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(37 78)(38 77)(39 76)(40 75)(41 74)(42 73)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 85)(58 96)(59 95)(60 94)
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), (1,19,58,43)(2,14,59,38)(3,21,60,45)(4,16,49,40)(5,23,50,47)(6,18,51,42)(7,13,52,37)(8,20,53,44)(9,15,54,39)(10,22,55,46)(11,17,56,41)(12,24,57,48)(25,72,79,85)(26,67,80,92)(27,62,81,87)(28,69,82,94)(29,64,83,89)(30,71,84,96)(31,66,73,91)(32,61,74,86)(33,68,75,93)(34,63,76,88)(35,70,77,95)(36,65,78,90), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,72)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,96)(59,95)(60,94)>;
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), (1,19,58,43)(2,14,59,38)(3,21,60,45)(4,16,49,40)(5,23,50,47)(6,18,51,42)(7,13,52,37)(8,20,53,44)(9,15,54,39)(10,22,55,46)(11,17,56,41)(12,24,57,48)(25,72,79,85)(26,67,80,92)(27,62,81,87)(28,69,82,94)(29,64,83,89)(30,71,84,96)(31,66,73,91)(32,61,74,86)(33,68,75,93)(34,63,76,88)(35,70,77,95)(36,65,78,90), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,72)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,96)(59,95)(60,94) );
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)], [(1,19,58,43),(2,14,59,38),(3,21,60,45),(4,16,49,40),(5,23,50,47),(6,18,51,42),(7,13,52,37),(8,20,53,44),(9,15,54,39),(10,22,55,46),(11,17,56,41),(12,24,57,48),(25,72,79,85),(26,67,80,92),(27,62,81,87),(28,69,82,94),(29,64,83,89),(30,71,84,96),(31,66,73,91),(32,61,74,86),(33,68,75,93),(34,63,76,88),(35,70,77,95),(36,65,78,90)], [(1,71),(2,70),(3,69),(4,68),(5,67),(6,66),(7,65),(8,64),(9,63),(10,62),(11,61),(12,72),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(37,78),(38,77),(39,76),(40,75),(41,74),(42,73),(43,84),(44,83),(45,82),(46,81),(47,80),(48,79),(49,93),(50,92),(51,91),(52,90),(53,89),(54,88),(55,87),(56,86),(57,85),(58,96),(59,95),(60,94)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 6T | 6U | 6V | 6W | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | 12AB | 12AC | 12AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
72 irreducible representations
| dim | 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 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | C4○D4 | C3×S3 | D12 | C3×D4 | C3×D4 | S3×C6 | C3×C4○D4 | C3×D12 | S3×D4 | Q8⋊3S3 | C3×S3×D4 | C3×Q8⋊3S3 |
| kernel | C3×C12⋊D4 | C3×D6⋊C4 | C32×C4⋊C4 | S3×C2×C12 | C6×D12 | C12⋊D4 | D6⋊C4 | C3×C4⋊C4 | S3×C2×C4 | C2×D12 | 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 | 1 | 1 | 3 | 2 | 4 | 2 | 2 | 6 | 1 | 2 | 2 | 3 | 2 | 2 | 4 | 4 | 4 | 6 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C12⋊D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 9 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 2 | 6 | 0 | 0 | 0 | 0 |
| 6 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,10,0,0,0,0,10,4,0,0,0,0,0,0,3,4,0,0,0,0,0,9,0,0,0,0,0,0,7,6,0,0,0,0,9,6],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,12],[2,6,0,0,0,0,6,11,0,0,0,0,0,0,12,0,0,0,0,0,5,1,0,0,0,0,0,0,1,0,0,0,0,0,2,12] >;
C3×C12⋊D4 in GAP, Magma, Sage, TeX
C_3\times C_{12}\rtimes D_4 % in TeX
G:=Group("C3xC12:D4"); // GroupNames label
G:=SmallGroup(288,666);
// by ID
G=gap.SmallGroup(288,666);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,142,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations