direct product, metabelian, supersoluble, monomial
Aliases: C2×D12.S3, D12.26D6, C12.22D12, C62.47D4, C3⋊C8⋊24D6, (C3×C6)⋊4SD16, C6⋊3(C24⋊C2), C6⋊1(D4.S3), (C2×D12).2S3, (C6×D12).7C2, (C2×C6).59D12, (C3×C12).66D4, C6.67(C2×D12), C32⋊7(C2×SD16), (C2×C12).117D6, C4.6(C3⋊D12), C12.71(C3⋊D4), C12.83(C22×S3), (C6×C12).77C22, (C3×C12).64C23, (C3×D12).34C22, C32⋊4Q8⋊15C22, C22.20(C3⋊D12), (C2×C3⋊C8)⋊6S3, (C6×C3⋊C8)⋊11C2, C4.52(C2×S32), (C2×C4).65S32, C3⋊4(C2×C24⋊C2), C3⋊1(C2×D4.S3), C6.3(C2×C3⋊D4), (C3×C3⋊C8)⋊29C22, (C3×C6).68(C2×D4), C2.7(C2×C3⋊D12), (C2×C32⋊4Q8)⋊9C2, (C2×C6).37(C3⋊D4), SmallGroup(288,476)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D12.S3
G = < a,b,c,d,e | a2=b12=c2=d3=1, e2=b9, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >
Subgroups: 610 in 155 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×SD16, C3⋊Dic3, C3×C12, S3×C6, C62, C24⋊C2, C2×C3⋊C8, D4.S3, C2×C24, C2×Dic6, C2×D12, C6×D4, C3×C3⋊C8, C3×D12, C3×D12, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C2×C24⋊C2, C2×D4.S3, D12.S3, C6×C3⋊C8, C6×D12, C2×C32⋊4Q8, C2×D12.S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C3⋊D4, C22×S3, C2×SD16, S32, C24⋊C2, D4.S3, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C2×C24⋊C2, C2×D4.S3, D12.S3, C2×C3⋊D12, C2×D12.S3
►On 96 points
Generators in S96
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 61)(59 62)(60 63)(73 94)(74 95)(75 96)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)
(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 63)(2 62)(3 61)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 49)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 75)(38 74)(39 73)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 78)(47 77)(48 76)
(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 53 57)(50 54 58)(51 55 59)(52 56 60)(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 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 29 10 26 7 35 4 32)(2 30 11 27 8 36 5 33)(3 31 12 28 9 25 6 34)(13 48 22 45 19 42 16 39)(14 37 23 46 20 43 17 40)(15 38 24 47 21 44 18 41)(49 79 58 76 55 73 52 82)(50 80 59 77 56 74 53 83)(51 81 60 78 57 75 54 84)(61 85 70 94 67 91 64 88)(62 86 71 95 68 92 65 89)(63 87 72 96 69 93 66 90)
G:=sub<Sym(96)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,61)(59,62)(60,63)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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), (1,63)(2,62)(3,61)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,49)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,75)(38,74)(39,73)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76), (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)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,29,10,26,7,35,4,32)(2,30,11,27,8,36,5,33)(3,31,12,28,9,25,6,34)(13,48,22,45,19,42,16,39)(14,37,23,46,20,43,17,40)(15,38,24,47,21,44,18,41)(49,79,58,76,55,73,52,82)(50,80,59,77,56,74,53,83)(51,81,60,78,57,75,54,84)(61,85,70,94,67,91,64,88)(62,86,71,95,68,92,65,89)(63,87,72,96,69,93,66,90)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,61)(59,62)(60,63)(73,94)(74,95)(75,96)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93), (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), (1,63)(2,62)(3,61)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,49)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,75)(38,74)(39,73)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76), (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)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,29,10,26,7,35,4,32)(2,30,11,27,8,36,5,33)(3,31,12,28,9,25,6,34)(13,48,22,45,19,42,16,39)(14,37,23,46,20,43,17,40)(15,38,24,47,21,44,18,41)(49,79,58,76,55,73,52,82)(50,80,59,77,56,74,53,83)(51,81,60,78,57,75,54,84)(61,85,70,94,67,91,64,88)(62,86,71,95,68,92,65,89)(63,87,72,96,69,93,66,90) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,13),(25,45),(26,46),(27,47),(28,48),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,61),(59,62),(60,63),(73,94),(74,95),(75,96),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91),(83,92),(84,93)], [(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)], [(1,63),(2,62),(3,61),(4,72),(5,71),(6,70),(7,69),(8,68),(9,67),(10,66),(11,65),(12,64),(13,49),(14,60),(15,59),(16,58),(17,57),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(25,88),(26,87),(27,86),(28,85),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,75),(38,74),(39,73),(40,84),(41,83),(42,82),(43,81),(44,80),(45,79),(46,78),(47,77),(48,76)], [(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),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,29,10,26,7,35,4,32),(2,30,11,27,8,36,5,33),(3,31,12,28,9,25,6,34),(13,48,22,45,19,42,16,39),(14,37,23,46,20,43,17,40),(15,38,24,47,21,44,18,41),(49,79,58,76,55,73,52,82),(50,80,59,77,56,74,53,83),(51,81,60,78,57,75,54,84),(61,85,70,94,67,91,64,88),(62,86,71,95,68,92,65,89),(63,87,72,96,69,93,66,90)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | D12 | C3⋊D4 | D12 | C3⋊D4 | C24⋊C2 | S32 | D4.S3 | C3⋊D12 | C2×S32 | C3⋊D12 | D12.S3 |
| kernel | C2×D12.S3 | D12.S3 | C6×C3⋊C8 | C6×D12 | C2×C32⋊4Q8 | C2×C3⋊C8 | C2×D12 | C3×C12 | C62 | C3⋊C8 | D12 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C2×C6 | C6 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 4 |
Matrix representation of C2×D12.S3 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 59 | 7 |
| 0 | 0 | 66 | 66 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 66 |
| 0 | 0 | 59 | 7 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 62 |
| 0 | 0 | 11 | 25 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,59,66,0,0,7,66],[1,0,0,0,0,1,0,0,0,0,66,59,0,0,66,7],[0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,36,11,0,0,62,25] >;
C2×D12.S3 in GAP, Magma, Sage, TeX
C_2\times D_{12}.S_3 % in TeX
G:=Group("C2xD12.S3"); // GroupNames label
G:=SmallGroup(288,476);
// by ID
G=gap.SmallGroup(288,476);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^3=1,e^2=b^9,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,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations