direct product, metabelian, supersoluble, monomial
Aliases: C3×D6.D4, C62.185C23, D6⋊C4⋊13C6, D6.5(C3×D4), C6.25(C6×D4), Dic3⋊C4⋊6C6, (C2×D12).3C6, (S3×C6).41D4, C6.184(S3×D4), (C6×D12).18C2, (C2×C12).272D6, C6.121(C4○D12), (C6×C12).193C22, C6.59(Q8⋊3S3), (C6×Dic3).128C22, C32⋊19(C22.D4), (C3×C4⋊C4)⋊5C6, C4⋊C4⋊2(C3×S3), (S3×C2×C4)⋊13C6, (C3×C4⋊C4)⋊11S3, C2.12(C3×S3×D4), (S3×C2×C12)⋊27C2, (C2×C4).9(S3×C6), (C3×D6⋊C4)⋊31C2, (C32×C4⋊C4)⋊6C2, (C2×C12).6(C2×C6), C6.11(C3×C4○D4), C22.49(S3×C2×C6), C2.14(C3×C4○D12), (C3×C6).213(C2×D4), (S3×C2×C6).94C22, C2.5(C3×Q8⋊3S3), (C3×Dic3⋊C4)⋊25C2, (C2×C6).40(C22×C6), C3⋊3(C3×C22.D4), (C3×C6).100(C4○D4), (C22×S3).21(C2×C6), (C2×C6).318(C22×S3), (C2×Dic3).29(C2×C6), SmallGroup(288,665)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D6.D4
G = < a,b,c,d,e | a3=b6=c2=d4=1, e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=d-1 >
Subgroups: 434 in 169 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22.D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×C12, C6×D4, S3×C12, C3×D12, C6×Dic3, C6×C12, S3×C2×C6, D6.D4, C3×C22.D4, C3×Dic3⋊C4, C3×D6⋊C4, C32×C4⋊C4, S3×C2×C12, C6×D12, C3×D6.D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3×D4, C22×S3, C22×C6, C22.D4, S3×C6, C4○D12, S3×D4, Q8⋊3S3, C6×D4, C3×C4○D4, S3×C2×C6, D6.D4, C3×C22.D4, C3×C4○D12, C3×S3×D4, C3×Q8⋊3S3, C3×D6.D4
►On 96 points
Generators in S96
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(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)(2 63)(3 62)(4 61)(5 66)(6 65)(7 44)(8 43)(9 48)(10 47)(11 46)(12 45)(13 56)(14 55)(15 60)(16 59)(17 58)(18 57)(19 75)(20 74)(21 73)(22 78)(23 77)(24 76)(25 68)(26 67)(27 72)(28 71)(29 70)(30 69)(31 87)(32 86)(33 85)(34 90)(35 89)(36 88)(37 80)(38 79)(39 84)(40 83)(41 82)(42 81)(49 91)(50 96)(51 95)(52 94)(53 93)(54 92)
(1 28 16 24)(2 29 17 19)(3 30 18 20)(4 25 13 21)(5 26 14 22)(6 27 15 23)(7 84 93 89)(8 79 94 90)(9 80 95 85)(10 81 96 86)(11 82 91 87)(12 83 92 88)(31 43 41 52)(32 44 42 53)(33 45 37 54)(34 46 38 49)(35 47 39 50)(36 48 40 51)(55 75 66 70)(56 76 61 71)(57 77 62 72)(58 78 63 67)(59 73 64 68)(60 74 65 69)
(1 54 4 51)(2 49 5 52)(3 50 6 53)(7 60 10 57)(8 55 11 58)(9 56 12 59)(13 48 16 45)(14 43 17 46)(15 44 18 47)(19 34 22 31)(20 35 23 32)(21 36 24 33)(25 40 28 37)(26 41 29 38)(27 42 30 39)(61 92 64 95)(62 93 65 96)(63 94 66 91)(67 79 70 82)(68 80 71 83)(69 81 72 84)(73 85 76 88)(74 86 77 89)(75 87 78 90)
G:=sub<Sym(96)| (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94), (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)(2,63)(3,62)(4,61)(5,66)(6,65)(7,44)(8,43)(9,48)(10,47)(11,46)(12,45)(13,56)(14,55)(15,60)(16,59)(17,58)(18,57)(19,75)(20,74)(21,73)(22,78)(23,77)(24,76)(25,68)(26,67)(27,72)(28,71)(29,70)(30,69)(31,87)(32,86)(33,85)(34,90)(35,89)(36,88)(37,80)(38,79)(39,84)(40,83)(41,82)(42,81)(49,91)(50,96)(51,95)(52,94)(53,93)(54,92), (1,28,16,24)(2,29,17,19)(3,30,18,20)(4,25,13,21)(5,26,14,22)(6,27,15,23)(7,84,93,89)(8,79,94,90)(9,80,95,85)(10,81,96,86)(11,82,91,87)(12,83,92,88)(31,43,41,52)(32,44,42,53)(33,45,37,54)(34,46,38,49)(35,47,39,50)(36,48,40,51)(55,75,66,70)(56,76,61,71)(57,77,62,72)(58,78,63,67)(59,73,64,68)(60,74,65,69), (1,54,4,51)(2,49,5,52)(3,50,6,53)(7,60,10,57)(8,55,11,58)(9,56,12,59)(13,48,16,45)(14,43,17,46)(15,44,18,47)(19,34,22,31)(20,35,23,32)(21,36,24,33)(25,40,28,37)(26,41,29,38)(27,42,30,39)(61,92,64,95)(62,93,65,96)(63,94,66,91)(67,79,70,82)(68,80,71,83)(69,81,72,84)(73,85,76,88)(74,86,77,89)(75,87,78,90)>;
G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94), (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)(2,63)(3,62)(4,61)(5,66)(6,65)(7,44)(8,43)(9,48)(10,47)(11,46)(12,45)(13,56)(14,55)(15,60)(16,59)(17,58)(18,57)(19,75)(20,74)(21,73)(22,78)(23,77)(24,76)(25,68)(26,67)(27,72)(28,71)(29,70)(30,69)(31,87)(32,86)(33,85)(34,90)(35,89)(36,88)(37,80)(38,79)(39,84)(40,83)(41,82)(42,81)(49,91)(50,96)(51,95)(52,94)(53,93)(54,92), (1,28,16,24)(2,29,17,19)(3,30,18,20)(4,25,13,21)(5,26,14,22)(6,27,15,23)(7,84,93,89)(8,79,94,90)(9,80,95,85)(10,81,96,86)(11,82,91,87)(12,83,92,88)(31,43,41,52)(32,44,42,53)(33,45,37,54)(34,46,38,49)(35,47,39,50)(36,48,40,51)(55,75,66,70)(56,76,61,71)(57,77,62,72)(58,78,63,67)(59,73,64,68)(60,74,65,69), (1,54,4,51)(2,49,5,52)(3,50,6,53)(7,60,10,57)(8,55,11,58)(9,56,12,59)(13,48,16,45)(14,43,17,46)(15,44,18,47)(19,34,22,31)(20,35,23,32)(21,36,24,33)(25,40,28,37)(26,41,29,38)(27,42,30,39)(61,92,64,95)(62,93,65,96)(63,94,66,91)(67,79,70,82)(68,80,71,83)(69,81,72,84)(73,85,76,88)(74,86,77,89)(75,87,78,90) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48),(49,51,53),(50,52,54),(55,59,57),(56,60,58),(61,65,63),(62,66,64),(67,71,69),(68,72,70),(73,77,75),(74,78,76),(79,83,81),(80,84,82),(85,89,87),(86,90,88),(91,95,93),(92,96,94)], [(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),(2,63),(3,62),(4,61),(5,66),(6,65),(7,44),(8,43),(9,48),(10,47),(11,46),(12,45),(13,56),(14,55),(15,60),(16,59),(17,58),(18,57),(19,75),(20,74),(21,73),(22,78),(23,77),(24,76),(25,68),(26,67),(27,72),(28,71),(29,70),(30,69),(31,87),(32,86),(33,85),(34,90),(35,89),(36,88),(37,80),(38,79),(39,84),(40,83),(41,82),(42,81),(49,91),(50,96),(51,95),(52,94),(53,93),(54,92)], [(1,28,16,24),(2,29,17,19),(3,30,18,20),(4,25,13,21),(5,26,14,22),(6,27,15,23),(7,84,93,89),(8,79,94,90),(9,80,95,85),(10,81,96,86),(11,82,91,87),(12,83,92,88),(31,43,41,52),(32,44,42,53),(33,45,37,54),(34,46,38,49),(35,47,39,50),(36,48,40,51),(55,75,66,70),(56,76,61,71),(57,77,62,72),(58,78,63,67),(59,73,64,68),(60,74,65,69)], [(1,54,4,51),(2,49,5,52),(3,50,6,53),(7,60,10,57),(8,55,11,58),(9,56,12,59),(13,48,16,45),(14,43,17,46),(15,44,18,47),(19,34,22,31),(20,35,23,32),(21,36,24,33),(25,40,28,37),(26,41,29,38),(27,42,30,39),(61,92,64,95),(62,93,65,96),(63,94,66,91),(67,79,70,82),(68,80,71,83),(69,81,72,84),(73,85,76,88),(74,86,77,89),(75,87,78,90)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 6T | 6U | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | 12AB | 12AC | 12AD | 12AE | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | C4○D4 | C3×S3 | C3×D4 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C4○D12 | S3×D4 | Q8⋊3S3 | C3×S3×D4 | C3×Q8⋊3S3 |
| kernel | C3×D6.D4 | C3×Dic3⋊C4 | C3×D6⋊C4 | C32×C4⋊C4 | S3×C2×C12 | C6×D12 | D6.D4 | Dic3⋊C4 | D6⋊C4 | C3×C4⋊C4 | S3×C2×C4 | C2×D12 | C3×C4⋊C4 | S3×C6 | C2×C12 | C3×C6 | C4⋊C4 | D6 | C2×C4 | C6 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 3 | 1 | 1 | 1 | 2 | 2 | 6 | 2 | 2 | 2 | 1 | 2 | 3 | 4 | 2 | 4 | 6 | 4 | 8 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×D6.D4 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 0 | 0 | 0 |
| 10 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 8 | 0 | 0 |
| 3 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 5 | 3 |
| 0 | 0 | 5 | 8 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[10,10,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,3,0,0,8,9,0,0,0,0,1,0,0,0,0,1],[1,12,0,0,0,12,0,0,0,0,5,0,0,0,3,8],[5,8,0,0,0,8,0,0,0,0,5,5,0,0,3,8] >;
C3×D6.D4 in GAP, Magma, Sage, TeX
C_3\times D_6.D_4
% in TeX
G:=Group("C3xD6.D4"); // GroupNames label
G:=SmallGroup(288,665);
// by ID
G=gap.SmallGroup(288,665);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,176,590,555,268,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^2=d^4=1,e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations