direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic3⋊D4, C24.37D6, D6⋊1(C2×D4), C6⋊1(C4⋊D4), C22⋊C4⋊39D6, Dic3⋊6(C2×D4), (C22×S3)⋊9D4, D6⋊C4⋊58C22, (C2×Dic3)⋊19D4, (C22×D12)⋊6C2, (C2×C6).33C24, C6.36(C22×D4), (C2×D12)⋊44C22, (C22×C4).186D6, C22.127(S3×D4), (C2×C12).573C23, Dic3⋊C4⋊48C22, (C22×S3).6C23, (C23×C6).59C22, C23.90(C22×S3), C22.72(S3×C23), C22.73(C4○D12), (S3×C23).96C22, (C22×C6).125C23, (C22×C12).353C22, (C2×Dic3).179C23, (C22×Dic3).77C22, C3⋊1(C2×C4⋊D4), C2.10(C2×S3×D4), (C2×D6⋊C4)⋊31C2, (S3×C22×C4)⋊18C2, (S3×C2×C4)⋊66C22, C6.13(C2×C4○D4), (C2×C22⋊C4)⋊12S3, (C6×C22⋊C4)⋊17C2, C2.15(C2×C4○D12), (C2×C6).382(C2×D4), (C22×C3⋊D4)⋊4C2, (C2×Dic3⋊C4)⋊22C2, (C2×C3⋊D4)⋊35C22, (C2×C6).102(C4○D4), (C3×C22⋊C4)⋊52C22, (C2×C4).134(C22×S3), SmallGroup(192,1048)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic3⋊D4
G = < a,b,c,d,e | a2=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b3c, ce=ec, ede=d-1 >
Subgroups: 1224 in 426 conjugacy classes, 127 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C2×C4⋊D4, Dic3⋊D4, C2×Dic3⋊C4, C2×D6⋊C4, C6×C22⋊C4, S3×C22×C4, C22×D12, C22×C3⋊D4, C2×Dic3⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C4⋊D4, C22×D4, C2×C4○D4, C4○D12, S3×D4, S3×C23, C2×C4⋊D4, Dic3⋊D4, C2×C4○D12, C2×S3×D4, C2×Dic3⋊D4
►On 96 points
Generators in S96
(1 60)(2 55)(3 56)(4 57)(5 58)(6 59)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(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 77 4 74)(2 76 5 73)(3 75 6 78)(7 35 10 32)(8 34 11 31)(9 33 12 36)(13 70 16 67)(14 69 17 72)(15 68 18 71)(19 61 22 64)(20 66 23 63)(21 65 24 62)(25 55 28 58)(26 60 29 57)(27 59 30 56)(37 94 40 91)(38 93 41 96)(39 92 42 95)(43 85 46 88)(44 90 47 87)(45 89 48 86)(49 79 52 82)(50 84 53 81)(51 83 54 80)
(1 84 14 86)(2 83 15 85)(3 82 16 90)(4 81 17 89)(5 80 18 88)(6 79 13 87)(7 20 91 28)(8 19 92 27)(9 24 93 26)(10 23 94 25)(11 22 95 30)(12 21 96 29)(31 61 39 59)(32 66 40 58)(33 65 41 57)(34 64 42 56)(35 63 37 55)(36 62 38 60)(43 76 51 68)(44 75 52 67)(45 74 53 72)(46 73 54 71)(47 78 49 70)(48 77 50 69)
(1 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 94)(14 93)(15 92)(16 91)(17 96)(18 95)(19 85)(20 90)(21 89)(22 88)(23 87)(24 86)(25 79)(26 84)(27 83)(28 82)(29 81)(30 80)(31 73)(32 78)(33 77)(34 76)(35 75)(36 74)(37 67)(38 72)(39 71)(40 70)(41 69)(42 68)(43 64)(44 63)(45 62)(46 61)(47 66)(48 65)(49 58)(50 57)(51 56)(52 55)(53 60)(54 59)
G:=sub<Sym(96)| (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (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), (1,77,4,74)(2,76,5,73)(3,75,6,78)(7,35,10,32)(8,34,11,31)(9,33,12,36)(13,70,16,67)(14,69,17,72)(15,68,18,71)(19,61,22,64)(20,66,23,63)(21,65,24,62)(25,55,28,58)(26,60,29,57)(27,59,30,56)(37,94,40,91)(38,93,41,96)(39,92,42,95)(43,85,46,88)(44,90,47,87)(45,89,48,86)(49,79,52,82)(50,84,53,81)(51,83,54,80), (1,84,14,86)(2,83,15,85)(3,82,16,90)(4,81,17,89)(5,80,18,88)(6,79,13,87)(7,20,91,28)(8,19,92,27)(9,24,93,26)(10,23,94,25)(11,22,95,30)(12,21,96,29)(31,61,39,59)(32,66,40,58)(33,65,41,57)(34,64,42,56)(35,63,37,55)(36,62,38,60)(43,76,51,68)(44,75,52,67)(45,74,53,72)(46,73,54,71)(47,78,49,70)(48,77,50,69), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,94)(14,93)(15,92)(16,91)(17,96)(18,95)(19,85)(20,90)(21,89)(22,88)(23,87)(24,86)(25,79)(26,84)(27,83)(28,82)(29,81)(30,80)(31,73)(32,78)(33,77)(34,76)(35,75)(36,74)(37,67)(38,72)(39,71)(40,70)(41,69)(42,68)(43,64)(44,63)(45,62)(46,61)(47,66)(48,65)(49,58)(50,57)(51,56)(52,55)(53,60)(54,59)>;
G:=Group( (1,60)(2,55)(3,56)(4,57)(5,58)(6,59)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (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), (1,77,4,74)(2,76,5,73)(3,75,6,78)(7,35,10,32)(8,34,11,31)(9,33,12,36)(13,70,16,67)(14,69,17,72)(15,68,18,71)(19,61,22,64)(20,66,23,63)(21,65,24,62)(25,55,28,58)(26,60,29,57)(27,59,30,56)(37,94,40,91)(38,93,41,96)(39,92,42,95)(43,85,46,88)(44,90,47,87)(45,89,48,86)(49,79,52,82)(50,84,53,81)(51,83,54,80), (1,84,14,86)(2,83,15,85)(3,82,16,90)(4,81,17,89)(5,80,18,88)(6,79,13,87)(7,20,91,28)(8,19,92,27)(9,24,93,26)(10,23,94,25)(11,22,95,30)(12,21,96,29)(31,61,39,59)(32,66,40,58)(33,65,41,57)(34,64,42,56)(35,63,37,55)(36,62,38,60)(43,76,51,68)(44,75,52,67)(45,74,53,72)(46,73,54,71)(47,78,49,70)(48,77,50,69), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,94)(14,93)(15,92)(16,91)(17,96)(18,95)(19,85)(20,90)(21,89)(22,88)(23,87)(24,86)(25,79)(26,84)(27,83)(28,82)(29,81)(30,80)(31,73)(32,78)(33,77)(34,76)(35,75)(36,74)(37,67)(38,72)(39,71)(40,70)(41,69)(42,68)(43,64)(44,63)(45,62)(46,61)(47,66)(48,65)(49,58)(50,57)(51,56)(52,55)(53,60)(54,59) );
G=PermutationGroup([[(1,60),(2,55),(3,56),(4,57),(5,58),(6,59),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(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)], [(1,77,4,74),(2,76,5,73),(3,75,6,78),(7,35,10,32),(8,34,11,31),(9,33,12,36),(13,70,16,67),(14,69,17,72),(15,68,18,71),(19,61,22,64),(20,66,23,63),(21,65,24,62),(25,55,28,58),(26,60,29,57),(27,59,30,56),(37,94,40,91),(38,93,41,96),(39,92,42,95),(43,85,46,88),(44,90,47,87),(45,89,48,86),(49,79,52,82),(50,84,53,81),(51,83,54,80)], [(1,84,14,86),(2,83,15,85),(3,82,16,90),(4,81,17,89),(5,80,18,88),(6,79,13,87),(7,20,91,28),(8,19,92,27),(9,24,93,26),(10,23,94,25),(11,22,95,30),(12,21,96,29),(31,61,39,59),(32,66,40,58),(33,65,41,57),(34,64,42,56),(35,63,37,55),(36,62,38,60),(43,76,51,68),(44,75,52,67),(45,74,53,72),(46,73,54,71),(47,78,49,70),(48,77,50,69)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,94),(14,93),(15,92),(16,91),(17,96),(18,95),(19,85),(20,90),(21,89),(22,88),(23,87),(24,86),(25,79),(26,84),(27,83),(28,82),(29,81),(30,80),(31,73),(32,78),(33,77),(34,76),(35,75),(36,74),(37,67),(38,72),(39,71),(40,70),(41,69),(42,68),(43,64),(44,63),(45,62),(46,61),(47,66),(48,65),(49,58),(50,57),(51,56),(52,55),(53,60),(54,59)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C4○D12 | S3×D4 |
| kernel | C2×Dic3⋊D4 | Dic3⋊D4 | C2×Dic3⋊C4 | C2×D6⋊C4 | C6×C22⋊C4 | S3×C22×C4 | C22×D12 | C22×C3⋊D4 | C2×C22⋊C4 | C2×Dic3 | C22×S3 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C22 | C22 |
| # reps | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 2 | 1 | 4 | 8 | 4 |
Matrix representation of C2×Dic3⋊D4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[12,0,0,0,0,0,2,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,10,0,0,0,0,3,6] >;
C2×Dic3⋊D4 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_3\rtimes D_4 % in TeX
G:=Group("C2xDic3:D4"); // GroupNames label
G:=SmallGroup(192,1048);
// by ID
G=gap.SmallGroup(192,1048);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,1571,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=d^4=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^3*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations