direct product, metabelian, supersoluble, monomial
Aliases: C2×Dic6⋊S3, Dic6⋊19D6, D12.31D6, C62.45D4, (C3×C6)⋊3SD16, C6⋊2(D4.S3), (C6×Dic6)⋊1C2, (C2×Dic6)⋊7S3, (C6×D12).6C2, (C2×D12).7S3, (C3×C12).64D4, C32⋊6(C2×SD16), C6⋊2(Q8⋊2S3), (C2×C12).115D6, C4.5(D6⋊S3), C12.45(C3⋊D4), (C6×C12).75C22, C12.81(C22×S3), (C3×C12).62C23, C32⋊4C8⋊19C22, (C3×Dic6)⋊20C22, (C3×D12).32C22, C22.13(D6⋊S3), C4.75(C2×S32), (C2×C4).109S32, C3⋊3(C2×D4.S3), C3⋊3(C2×Q8⋊2S3), (C3×C6).66(C2×D4), C6.72(C2×C3⋊D4), (C2×C32⋊4C8)⋊7C2, C2.7(C2×D6⋊S3), (C2×C6).56(C3⋊D4), SmallGroup(288,474)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic6⋊S3
G = < a,b,c,d,e | a2=b12=d3=e2=1, c2=b6, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b7, cd=dc, ece=b3c, ede=d-1 >
Subgroups: 498 in 147 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, Dic6, Dic6, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C2×SD16, C3×Dic3, C3×C12, S3×C6, C62, C2×C3⋊C8, D4.S3, Q8⋊2S3, C2×Dic6, C2×D12, C6×D4, C6×Q8, C32⋊4C8, C3×Dic6, C3×Dic6, C3×D12, C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C2×D4.S3, C2×Q8⋊2S3, Dic6⋊S3, C2×C32⋊4C8, C6×Dic6, C6×D12, C2×Dic6⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C2×SD16, S32, D4.S3, Q8⋊2S3, C2×C3⋊D4, D6⋊S3, C2×S32, C2×D4.S3, C2×Q8⋊2S3, Dic6⋊S3, C2×D6⋊S3, C2×Dic6⋊S3
►On 96 points
Generators in S96
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)
(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 30 7 36)(2 29 8 35)(3 28 9 34)(4 27 10 33)(5 26 11 32)(6 25 12 31)(13 37 19 43)(14 48 20 42)(15 47 21 41)(16 46 22 40)(17 45 23 39)(18 44 24 38)(49 77 55 83)(50 76 56 82)(51 75 57 81)(52 74 58 80)(53 73 59 79)(54 84 60 78)(61 95 67 89)(62 94 68 88)(63 93 69 87)(64 92 70 86)(65 91 71 85)(66 90 72 96)
(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 64)(2 71)(3 66)(4 61)(5 68)(6 63)(7 70)(8 65)(9 72)(10 67)(11 62)(12 69)(13 58)(14 53)(15 60)(16 55)(17 50)(18 57)(19 52)(20 59)(21 54)(22 49)(23 56)(24 51)(25 90)(26 85)(27 92)(28 87)(29 94)(30 89)(31 96)(32 91)(33 86)(34 93)(35 88)(36 95)(37 77)(38 84)(39 79)(40 74)(41 81)(42 76)(43 83)(44 78)(45 73)(46 80)(47 75)(48 82)
G:=sub<Sym(96)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (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), (1,30,7,36)(2,29,8,35)(3,28,9,34)(4,27,10,33)(5,26,11,32)(6,25,12,31)(13,37,19,43)(14,48,20,42)(15,47,21,41)(16,46,22,40)(17,45,23,39)(18,44,24,38)(49,77,55,83)(50,76,56,82)(51,75,57,81)(52,74,58,80)(53,73,59,79)(54,84,60,78)(61,95,67,89)(62,94,68,88)(63,93,69,87)(64,92,70,86)(65,91,71,85)(66,90,72,96), (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), (1,64)(2,71)(3,66)(4,61)(5,68)(6,63)(7,70)(8,65)(9,72)(10,67)(11,62)(12,69)(13,58)(14,53)(15,60)(16,55)(17,50)(18,57)(19,52)(20,59)(21,54)(22,49)(23,56)(24,51)(25,90)(26,85)(27,92)(28,87)(29,94)(30,89)(31,96)(32,91)(33,86)(34,93)(35,88)(36,95)(37,77)(38,84)(39,79)(40,74)(41,81)(42,76)(43,83)(44,78)(45,73)(46,80)(47,75)(48,82)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (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), (1,30,7,36)(2,29,8,35)(3,28,9,34)(4,27,10,33)(5,26,11,32)(6,25,12,31)(13,37,19,43)(14,48,20,42)(15,47,21,41)(16,46,22,40)(17,45,23,39)(18,44,24,38)(49,77,55,83)(50,76,56,82)(51,75,57,81)(52,74,58,80)(53,73,59,79)(54,84,60,78)(61,95,67,89)(62,94,68,88)(63,93,69,87)(64,92,70,86)(65,91,71,85)(66,90,72,96), (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), (1,64)(2,71)(3,66)(4,61)(5,68)(6,63)(7,70)(8,65)(9,72)(10,67)(11,62)(12,69)(13,58)(14,53)(15,60)(16,55)(17,50)(18,57)(19,52)(20,59)(21,54)(22,49)(23,56)(24,51)(25,90)(26,85)(27,92)(28,87)(29,94)(30,89)(31,96)(32,91)(33,86)(34,93)(35,88)(36,95)(37,77)(38,84)(39,79)(40,74)(41,81)(42,76)(43,83)(44,78)(45,73)(46,80)(47,75)(48,82) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,37),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90)], [(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)], [(1,30,7,36),(2,29,8,35),(3,28,9,34),(4,27,10,33),(5,26,11,32),(6,25,12,31),(13,37,19,43),(14,48,20,42),(15,47,21,41),(16,46,22,40),(17,45,23,39),(18,44,24,38),(49,77,55,83),(50,76,56,82),(51,75,57,81),(52,74,58,80),(53,73,59,79),(54,84,60,78),(61,95,67,89),(62,94,68,88),(63,93,69,87),(64,92,70,86),(65,91,71,85),(66,90,72,96)], [(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)], [(1,64),(2,71),(3,66),(4,61),(5,68),(6,63),(7,70),(8,65),(9,72),(10,67),(11,62),(12,69),(13,58),(14,53),(15,60),(16,55),(17,50),(18,57),(19,52),(20,59),(21,54),(22,49),(23,56),(24,51),(25,90),(26,85),(27,92),(28,87),(29,94),(30,89),(31,96),(32,91),(33,86),(34,93),(35,88),(36,95),(37,77),(38,84),(39,79),(40,74),(41,81),(42,76),(43,83),(44,78),(45,73),(46,80),(47,75),(48,82)]])
42 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 | ··· | 12H | 12I | 12J | 12K | 12L |
| 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 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | C3⋊D4 | C3⋊D4 | S32 | D4.S3 | Q8⋊2S3 | D6⋊S3 | C2×S32 | D6⋊S3 | Dic6⋊S3 |
| kernel | C2×Dic6⋊S3 | Dic6⋊S3 | C2×C32⋊4C8 | C6×Dic6 | C6×D12 | C2×Dic6 | C2×D12 | C3×C12 | C62 | Dic6 | D12 | C2×C12 | C3×C6 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 1 | 1 | 4 |
Matrix representation of C2×Dic6⋊S3 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 19 | 21 | 0 | 0 | 0 | 0 |
| 21 | 54 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 52 | 61 | 0 | 0 | 0 | 0 |
| 61 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 5 |
| 0 | 0 | 0 | 0 | 55 | 23 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[19,21,0,0,0,0,21,54,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,30,60,0,0,0,0,13,43],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[52,61,0,0,0,0,61,21,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,50,55,0,0,0,0,5,23] >;
C2×Dic6⋊S3 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_6\rtimes S_3 % in TeX
G:=Group("C2xDic6:S3"); // GroupNames label
G:=SmallGroup(288,474);
// by ID
G=gap.SmallGroup(288,474);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^12=d^3=e^2=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b^7,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations