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, C22.72(S3×C23), C23.90(C22×S3), (C23×C6).59C22, 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
Subgroups: 1224 in 426 conjugacy classes, 127 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×10], C22, C22 [×6], C22 [×36], S3 [×6], C6 [×3], C6 [×4], C6 [×2], C2×C4 [×4], C2×C4 [×22], D4 [×24], C23, C23 [×2], C23 [×24], Dic3 [×4], Dic3 [×2], C12 [×4], D6 [×4], D6 [×22], C2×C6, C2×C6 [×6], C2×C6 [×10], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×10], C2×D4 [×24], C24, C24 [×2], C4×S3 [×8], D12 [×8], C2×Dic3 [×8], C2×Dic3 [×2], C3⋊D4 [×16], C2×C12 [×4], C2×C12 [×4], C22×S3 [×8], C22×S3 [×10], C22×C6, C22×C6 [×2], C22×C6 [×6], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], Dic3⋊C4 [×4], D6⋊C4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], S3×C2×C4 [×4], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×8], C22×C12 [×2], S3×C23 [×2], C23×C6, C2×C4⋊D4, Dic3⋊D4 [×8], C2×Dic3⋊C4, C2×D6⋊C4, C6×C22⋊C4, S3×C22×C4, C22×D12, C22×C3⋊D4 [×2], C2×Dic3⋊D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C4○D4 [×2], C24, C22×S3 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C4○D12 [×2], S3×D4 [×4], S3×C23, C2×C4⋊D4, Dic3⋊D4 [×4], C2×C4○D12, C2×S3×D4 [×2], C2×Dic3⋊D4
Generators and relations
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 >
►On 96 points
(1 58)(2 59)(3 60)(4 55)(5 56)(6 57)(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 33 10 36)(8 32 11 35)(9 31 12 34)(13 68 16 71)(14 67 17 70)(15 72 18 69)(19 65 22 62)(20 64 23 61)(21 63 24 66)(25 59 28 56)(26 58 29 55)(27 57 30 60)(37 92 40 95)(38 91 41 94)(39 96 42 93)(43 89 46 86)(44 88 47 85)(45 87 48 90)(49 83 52 80)(50 82 53 79)(51 81 54 84)
(1 82 14 86)(2 81 15 85)(3 80 16 90)(4 79 17 89)(5 84 18 88)(6 83 13 87)(7 24 95 28)(8 23 96 27)(9 22 91 26)(10 21 92 25)(11 20 93 30)(12 19 94 29)(31 65 41 55)(32 64 42 60)(33 63 37 59)(34 62 38 58)(35 61 39 57)(36 66 40 56)(43 74 53 70)(44 73 54 69)(45 78 49 68)(46 77 50 67)(47 76 51 72)(48 75 52 71)
(1 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 92)(14 91)(15 96)(16 95)(17 94)(18 93)(19 89)(20 88)(21 87)(22 86)(23 85)(24 90)(25 83)(26 82)(27 81)(28 80)(29 79)(30 84)(31 77)(32 76)(33 75)(34 74)(35 73)(36 78)(37 71)(38 70)(39 69)(40 68)(41 67)(42 72)(43 62)(44 61)(45 66)(46 65)(47 64)(48 63)(49 56)(50 55)(51 60)(52 59)(53 58)(54 57)
G:=sub<Sym(96)| (1,58)(2,59)(3,60)(4,55)(5,56)(6,57)(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)(7,33,10,36)(8,32,11,35)(9,31,12,34)(13,68,16,71)(14,67,17,70)(15,72,18,69)(19,65,22,62)(20,64,23,61)(21,63,24,66)(25,59,28,56)(26,58,29,55)(27,57,30,60)(37,92,40,95)(38,91,41,94)(39,96,42,93)(43,89,46,86)(44,88,47,85)(45,87,48,90)(49,83,52,80)(50,82,53,79)(51,81,54,84), (1,82,14,86)(2,81,15,85)(3,80,16,90)(4,79,17,89)(5,84,18,88)(6,83,13,87)(7,24,95,28)(8,23,96,27)(9,22,91,26)(10,21,92,25)(11,20,93,30)(12,19,94,29)(31,65,41,55)(32,64,42,60)(33,63,37,59)(34,62,38,58)(35,61,39,57)(36,66,40,56)(43,74,53,70)(44,73,54,69)(45,78,49,68)(46,77,50,67)(47,76,51,72)(48,75,52,71), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,92)(14,91)(15,96)(16,95)(17,94)(18,93)(19,89)(20,88)(21,87)(22,86)(23,85)(24,90)(25,83)(26,82)(27,81)(28,80)(29,79)(30,84)(31,77)(32,76)(33,75)(34,74)(35,73)(36,78)(37,71)(38,70)(39,69)(40,68)(41,67)(42,72)(43,62)(44,61)(45,66)(46,65)(47,64)(48,63)(49,56)(50,55)(51,60)(52,59)(53,58)(54,57)>;
G:=Group( (1,58)(2,59)(3,60)(4,55)(5,56)(6,57)(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)(7,33,10,36)(8,32,11,35)(9,31,12,34)(13,68,16,71)(14,67,17,70)(15,72,18,69)(19,65,22,62)(20,64,23,61)(21,63,24,66)(25,59,28,56)(26,58,29,55)(27,57,30,60)(37,92,40,95)(38,91,41,94)(39,96,42,93)(43,89,46,86)(44,88,47,85)(45,87,48,90)(49,83,52,80)(50,82,53,79)(51,81,54,84), (1,82,14,86)(2,81,15,85)(3,80,16,90)(4,79,17,89)(5,84,18,88)(6,83,13,87)(7,24,95,28)(8,23,96,27)(9,22,91,26)(10,21,92,25)(11,20,93,30)(12,19,94,29)(31,65,41,55)(32,64,42,60)(33,63,37,59)(34,62,38,58)(35,61,39,57)(36,66,40,56)(43,74,53,70)(44,73,54,69)(45,78,49,68)(46,77,50,67)(47,76,51,72)(48,75,52,71), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,92)(14,91)(15,96)(16,95)(17,94)(18,93)(19,89)(20,88)(21,87)(22,86)(23,85)(24,90)(25,83)(26,82)(27,81)(28,80)(29,79)(30,84)(31,77)(32,76)(33,75)(34,74)(35,73)(36,78)(37,71)(38,70)(39,69)(40,68)(41,67)(42,72)(43,62)(44,61)(45,66)(46,65)(47,64)(48,63)(49,56)(50,55)(51,60)(52,59)(53,58)(54,57) );
G=PermutationGroup([(1,58),(2,59),(3,60),(4,55),(5,56),(6,57),(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),(7,33,10,36),(8,32,11,35),(9,31,12,34),(13,68,16,71),(14,67,17,70),(15,72,18,69),(19,65,22,62),(20,64,23,61),(21,63,24,66),(25,59,28,56),(26,58,29,55),(27,57,30,60),(37,92,40,95),(38,91,41,94),(39,96,42,93),(43,89,46,86),(44,88,47,85),(45,87,48,90),(49,83,52,80),(50,82,53,79),(51,81,54,84)], [(1,82,14,86),(2,81,15,85),(3,80,16,90),(4,79,17,89),(5,84,18,88),(6,83,13,87),(7,24,95,28),(8,23,96,27),(9,22,91,26),(10,21,92,25),(11,20,93,30),(12,19,94,29),(31,65,41,55),(32,64,42,60),(33,63,37,59),(34,62,38,58),(35,61,39,57),(36,66,40,56),(43,74,53,70),(44,73,54,69),(45,78,49,68),(46,77,50,67),(47,76,51,72),(48,75,52,71)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,92),(14,91),(15,96),(16,95),(17,94),(18,93),(19,89),(20,88),(21,87),(22,86),(23,85),(24,90),(25,83),(26,82),(27,81),(28,80),(29,79),(30,84),(31,77),(32,76),(33,75),(34,74),(35,73),(36,78),(37,71),(38,70),(39,69),(40,68),(41,67),(42,72),(43,62),(44,61),(45,66),(46,65),(47,64),(48,63),(49,56),(50,55),(51,60),(52,59),(53,58),(54,57)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
C_2\times 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