metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊10D4, C42.142D6, C6.912- (1+4), C4.71(S3×D4), (C4×D12)⋊44C2, C3⋊5(Q8⋊5D4), C12.64(C2×D4), D6⋊15(C4○D4), D6⋊3D4⋊34C2, Dic3⋊D4⋊41C2, D6⋊3Q8⋊29C2, (C4×Dic6)⋊45C2, (C2×D4).174D6, C4.4D4⋊11S3, (C2×Q8).161D6, C22⋊C4.73D6, C6.91(C22×D4), (C2×C6).221C24, (C2×C12).81C23, C2.52(Q8○D12), Dic3.28(C2×D4), Dic3⋊4D4⋊30C2, (C4×C12).186C22, D6⋊C4.135C22, (C6×D4).156C22, C23.53(C22×S3), (C22×C6).51C23, (C6×Q8).127C22, C23.11D6⋊40C2, Dic3.D4⋊41C2, (C2×D12).223C22, C4⋊Dic3.377C22, C22.242(S3×C23), Dic3⋊C4.121C22, (C22×S3).216C23, (C2×Dic6).297C22, (C4×Dic3).214C22, (C2×Dic3).253C23, C6.D4.55C22, (C22×Dic3).143C22, (C2×S3×Q8)⋊11C2, C2.64(C2×S3×D4), C2.77(S3×C4○D4), C6.188(C2×C4○D4), (C2×D4⋊2S3)⋊19C2, (C3×C4.4D4)⋊13C2, (S3×C2×C4).121C22, (C2×C4).196(C22×S3), (C2×C3⋊D4).60C22, (C3×C22⋊C4).65C22, SmallGroup(192,1236)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 736 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×2], C4 [×12], C22, C22 [×13], S3 [×3], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×12], Q8 [×10], C23 [×2], C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×2], C12 [×4], D6 [×2], D6 [×5], C2×C6, C2×C6 [×6], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic6 [×4], Dic6 [×4], C4×S3 [×8], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×C6 [×2], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4, C4.4D4 [×2], C22×Q8, C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×2], D6⋊C4 [×2], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], C2×Dic6, C2×Dic6 [×2], S3×C2×C4 [×2], S3×C2×C4 [×2], C2×D12, D4⋊2S3 [×4], S3×Q8 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×4], C6×D4, C6×Q8, Q8⋊5D4, C4×Dic6, C4×D12, Dic3.D4 [×2], Dic3⋊4D4 [×2], Dic3⋊D4 [×2], C23.11D6 [×2], D6⋊3D4, D6⋊3Q8, C3×C4.4D4, C2×D4⋊2S3, C2×S3×Q8, Dic6⋊10D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2- (1+4), S3×D4 [×2], S3×C23, Q8⋊5D4, C2×S3×D4, S3×C4○D4, Q8○D12, Dic6⋊10D4
Generators and relations
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a5, cbc-1=dbd=a6b, dcd=c-1 >
►On 96 points
(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 36 7 30)(2 35 8 29)(3 34 9 28)(4 33 10 27)(5 32 11 26)(6 31 12 25)(13 65 19 71)(14 64 20 70)(15 63 21 69)(16 62 22 68)(17 61 23 67)(18 72 24 66)(37 94 43 88)(38 93 44 87)(39 92 45 86)(40 91 46 85)(41 90 47 96)(42 89 48 95)(49 81 55 75)(50 80 56 74)(51 79 57 73)(52 78 58 84)(53 77 59 83)(54 76 60 82)
(1 92 63 49)(2 93 64 50)(3 94 65 51)(4 95 66 52)(5 96 67 53)(6 85 68 54)(7 86 69 55)(8 87 70 56)(9 88 71 57)(10 89 72 58)(11 90 61 59)(12 91 62 60)(13 79 28 43)(14 80 29 44)(15 81 30 45)(16 82 31 46)(17 83 32 47)(18 84 33 48)(19 73 34 37)(20 74 35 38)(21 75 36 39)(22 76 25 40)(23 77 26 41)(24 78 27 42)
(1 63)(2 68)(3 61)(4 66)(5 71)(6 64)(7 69)(8 62)(9 67)(10 72)(11 65)(12 70)(13 26)(14 31)(15 36)(16 29)(17 34)(18 27)(19 32)(20 25)(21 30)(22 35)(23 28)(24 33)(37 47)(38 40)(39 45)(41 43)(42 48)(44 46)(50 54)(51 59)(53 57)(56 60)(73 83)(74 76)(75 81)(77 79)(78 84)(80 82)(85 93)(87 91)(88 96)(90 94)
G:=sub<Sym(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), (1,36,7,30)(2,35,8,29)(3,34,9,28)(4,33,10,27)(5,32,11,26)(6,31,12,25)(13,65,19,71)(14,64,20,70)(15,63,21,69)(16,62,22,68)(17,61,23,67)(18,72,24,66)(37,94,43,88)(38,93,44,87)(39,92,45,86)(40,91,46,85)(41,90,47,96)(42,89,48,95)(49,81,55,75)(50,80,56,74)(51,79,57,73)(52,78,58,84)(53,77,59,83)(54,76,60,82), (1,92,63,49)(2,93,64,50)(3,94,65,51)(4,95,66,52)(5,96,67,53)(6,85,68,54)(7,86,69,55)(8,87,70,56)(9,88,71,57)(10,89,72,58)(11,90,61,59)(12,91,62,60)(13,79,28,43)(14,80,29,44)(15,81,30,45)(16,82,31,46)(17,83,32,47)(18,84,33,48)(19,73,34,37)(20,74,35,38)(21,75,36,39)(22,76,25,40)(23,77,26,41)(24,78,27,42), (1,63)(2,68)(3,61)(4,66)(5,71)(6,64)(7,69)(8,62)(9,67)(10,72)(11,65)(12,70)(13,26)(14,31)(15,36)(16,29)(17,34)(18,27)(19,32)(20,25)(21,30)(22,35)(23,28)(24,33)(37,47)(38,40)(39,45)(41,43)(42,48)(44,46)(50,54)(51,59)(53,57)(56,60)(73,83)(74,76)(75,81)(77,79)(78,84)(80,82)(85,93)(87,91)(88,96)(90,94)>;
G:=Group( (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), (1,36,7,30)(2,35,8,29)(3,34,9,28)(4,33,10,27)(5,32,11,26)(6,31,12,25)(13,65,19,71)(14,64,20,70)(15,63,21,69)(16,62,22,68)(17,61,23,67)(18,72,24,66)(37,94,43,88)(38,93,44,87)(39,92,45,86)(40,91,46,85)(41,90,47,96)(42,89,48,95)(49,81,55,75)(50,80,56,74)(51,79,57,73)(52,78,58,84)(53,77,59,83)(54,76,60,82), (1,92,63,49)(2,93,64,50)(3,94,65,51)(4,95,66,52)(5,96,67,53)(6,85,68,54)(7,86,69,55)(8,87,70,56)(9,88,71,57)(10,89,72,58)(11,90,61,59)(12,91,62,60)(13,79,28,43)(14,80,29,44)(15,81,30,45)(16,82,31,46)(17,83,32,47)(18,84,33,48)(19,73,34,37)(20,74,35,38)(21,75,36,39)(22,76,25,40)(23,77,26,41)(24,78,27,42), (1,63)(2,68)(3,61)(4,66)(5,71)(6,64)(7,69)(8,62)(9,67)(10,72)(11,65)(12,70)(13,26)(14,31)(15,36)(16,29)(17,34)(18,27)(19,32)(20,25)(21,30)(22,35)(23,28)(24,33)(37,47)(38,40)(39,45)(41,43)(42,48)(44,46)(50,54)(51,59)(53,57)(56,60)(73,83)(74,76)(75,81)(77,79)(78,84)(80,82)(85,93)(87,91)(88,96)(90,94) );
G=PermutationGroup([(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)], [(1,36,7,30),(2,35,8,29),(3,34,9,28),(4,33,10,27),(5,32,11,26),(6,31,12,25),(13,65,19,71),(14,64,20,70),(15,63,21,69),(16,62,22,68),(17,61,23,67),(18,72,24,66),(37,94,43,88),(38,93,44,87),(39,92,45,86),(40,91,46,85),(41,90,47,96),(42,89,48,95),(49,81,55,75),(50,80,56,74),(51,79,57,73),(52,78,58,84),(53,77,59,83),(54,76,60,82)], [(1,92,63,49),(2,93,64,50),(3,94,65,51),(4,95,66,52),(5,96,67,53),(6,85,68,54),(7,86,69,55),(8,87,70,56),(9,88,71,57),(10,89,72,58),(11,90,61,59),(12,91,62,60),(13,79,28,43),(14,80,29,44),(15,81,30,45),(16,82,31,46),(17,83,32,47),(18,84,33,48),(19,73,34,37),(20,74,35,38),(21,75,36,39),(22,76,25,40),(23,77,26,41),(24,78,27,42)], [(1,63),(2,68),(3,61),(4,66),(5,71),(6,64),(7,69),(8,62),(9,67),(10,72),(11,65),(12,70),(13,26),(14,31),(15,36),(16,29),(17,34),(18,27),(19,32),(20,25),(21,30),(22,35),(23,28),(24,33),(37,47),(38,40),(39,45),(41,43),(42,48),(44,46),(50,54),(51,59),(53,57),(56,60),(73,83),(74,76),(75,81),(77,79),(78,84),(80,82),(85,93),(87,91),(88,96),(90,94)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 1 | 2 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,2,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 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 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4○D4 | 2- (1+4) | S3×D4 | S3×C4○D4 | Q8○D12 |
| kernel | Dic6⋊10D4 | C4×Dic6 | C4×D12 | Dic3.D4 | Dic3⋊4D4 | Dic3⋊D4 | C23.11D6 | D6⋊3D4 | D6⋊3Q8 | C3×C4.4D4 | C2×D4⋊2S3 | C2×S3×Q8 | C4.4D4 | Dic6 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_{10}D_4 % in TeX
G:=Group("Dic6:10D4"); // GroupNames label
G:=SmallGroup(192,1236);
// by ID
G=gap.SmallGroup(192,1236);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,1571,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,c*b*c^-1=d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations