metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊21D4, C6.762- (1+4), C3⋊3(D4×Q8), D6⋊6(C2×Q8), C3⋊D4⋊1Q8, C12⋊Q8⋊25C2, C22⋊Q8⋊9S3, C22⋊2(S3×Q8), C4⋊C4.190D6, Dic3⋊4(C2×Q8), C4.113(S3×D4), D6⋊Q8⋊19C2, C4.D12⋊26C2, C12.236(C2×D4), (C2×Q8).151D6, C22⋊C4.58D6, C6.78(C22×D4), C6.35(C22×Q8), (C2×C6).176C24, (C2×C12).55C23, C2.36(Q8○D12), Dic3.24(C2×D4), (C22×C4).254D6, Dic3⋊Q8⋊15C2, Dic6⋊C4⋊25C2, D6⋊C4.107C22, Dic3⋊4D4.1C2, (C22×Dic6)⋊17C2, (C6×Q8).108C22, Dic3.D4⋊23C2, Dic3⋊C4.28C22, C4⋊Dic3.216C22, C23.200(C22×S3), C22.197(S3×C23), (C22×C6).204C23, (C22×S3).198C23, (C22×C12).256C22, (C4×Dic3).106C22, (C2×Dic6).248C22, (C2×Dic3).235C23, C6.D4.117C22, (C22×Dic3).118C22, (C2×S3×Q8)⋊7C2, (C2×C6)⋊3(C2×Q8), C2.51(C2×S3×D4), C2.18(C2×S3×Q8), (C4×C3⋊D4).7C2, (S3×C2×C4).96C22, (C3×C22⋊Q8)⋊12C2, (C2×C4).49(C22×S3), (C3×C4⋊C4).159C22, (C2×C3⋊D4).124C22, (C3×C22⋊C4).31C22, SmallGroup(192,1191)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 656 in 280 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×15], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×4], Q8 [×16], C23, C23, Dic3 [×6], Dic3 [×4], C12 [×2], C12 [×5], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4, C2×Q8, C2×Q8 [×14], Dic6 [×4], Dic6 [×10], C4×S3 [×6], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C3×Q8 [×2], C22×S3, C22×C6, C4×D4 [×3], C4×Q8, C22⋊Q8, C22⋊Q8 [×5], C4⋊Q8 [×3], C22×Q8 [×2], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4 [×6], C4⋊Dic3 [×2], D6⋊C4, D6⋊C4 [×2], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6 [×2], C2×Dic6 [×4], C2×Dic6 [×4], S3×C2×C4, S3×C2×C4 [×2], S3×Q8 [×4], C22×Dic3 [×2], C2×C3⋊D4, C22×C12, C6×Q8, D4×Q8, Dic3.D4 [×2], Dic3⋊4D4 [×2], Dic6⋊C4, C12⋊Q8 [×2], D6⋊Q8 [×2], C4.D12, C4×C3⋊D4, Dic3⋊Q8, C3×C22⋊Q8, C22×Dic6, C2×S3×Q8, Dic6⋊21D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C24, C22×S3 [×7], C22×D4, C22×Q8, 2- (1+4), S3×D4 [×2], S3×Q8 [×2], S3×C23, D4×Q8, C2×S3×D4, C2×S3×Q8, Q8○D12, Dic6⋊21D4
Generators and relations
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a5, ad=da, bc=cb, bd=db, 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 31 7 25)(2 30 8 36)(3 29 9 35)(4 28 10 34)(5 27 11 33)(6 26 12 32)(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 93 43 87)(38 92 44 86)(39 91 45 85)(40 90 46 96)(41 89 47 95)(42 88 48 94)(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 91 55 62)(2 96 56 67)(3 89 57 72)(4 94 58 65)(5 87 59 70)(6 92 60 63)(7 85 49 68)(8 90 50 61)(9 95 51 66)(10 88 52 71)(11 93 53 64)(12 86 54 69)(13 34 48 78)(14 27 37 83)(15 32 38 76)(16 25 39 81)(17 30 40 74)(18 35 41 79)(19 28 42 84)(20 33 43 77)(21 26 44 82)(22 31 45 75)(23 36 46 80)(24 29 47 73)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 61)(13 34)(14 35)(15 36)(16 25)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)
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), (1,31,7,25)(2,30,8,36)(3,29,9,35)(4,28,10,34)(5,27,11,33)(6,26,12,32)(13,65,19,71)(14,64,20,70)(15,63,21,69)(16,62,22,68)(17,61,23,67)(18,72,24,66)(37,93,43,87)(38,92,44,86)(39,91,45,85)(40,90,46,96)(41,89,47,95)(42,88,48,94)(49,81,55,75)(50,80,56,74)(51,79,57,73)(52,78,58,84)(53,77,59,83)(54,76,60,82), (1,91,55,62)(2,96,56,67)(3,89,57,72)(4,94,58,65)(5,87,59,70)(6,92,60,63)(7,85,49,68)(8,90,50,61)(9,95,51,66)(10,88,52,71)(11,93,53,64)(12,86,54,69)(13,34,48,78)(14,27,37,83)(15,32,38,76)(16,25,39,81)(17,30,40,74)(18,35,41,79)(19,28,42,84)(20,33,43,77)(21,26,44,82)(22,31,45,75)(23,36,46,80)(24,29,47,73), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,34)(14,35)(15,36)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)>;
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), (1,31,7,25)(2,30,8,36)(3,29,9,35)(4,28,10,34)(5,27,11,33)(6,26,12,32)(13,65,19,71)(14,64,20,70)(15,63,21,69)(16,62,22,68)(17,61,23,67)(18,72,24,66)(37,93,43,87)(38,92,44,86)(39,91,45,85)(40,90,46,96)(41,89,47,95)(42,88,48,94)(49,81,55,75)(50,80,56,74)(51,79,57,73)(52,78,58,84)(53,77,59,83)(54,76,60,82), (1,91,55,62)(2,96,56,67)(3,89,57,72)(4,94,58,65)(5,87,59,70)(6,92,60,63)(7,85,49,68)(8,90,50,61)(9,95,51,66)(10,88,52,71)(11,93,53,64)(12,86,54,69)(13,34,48,78)(14,27,37,83)(15,32,38,76)(16,25,39,81)(17,30,40,74)(18,35,41,79)(19,28,42,84)(20,33,43,77)(21,26,44,82)(22,31,45,75)(23,36,46,80)(24,29,47,73), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,34)(14,35)(15,36)(16,25)(17,26)(18,27)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96) );
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)], [(1,31,7,25),(2,30,8,36),(3,29,9,35),(4,28,10,34),(5,27,11,33),(6,26,12,32),(13,65,19,71),(14,64,20,70),(15,63,21,69),(16,62,22,68),(17,61,23,67),(18,72,24,66),(37,93,43,87),(38,92,44,86),(39,91,45,85),(40,90,46,96),(41,89,47,95),(42,88,48,94),(49,81,55,75),(50,80,56,74),(51,79,57,73),(52,78,58,84),(53,77,59,83),(54,76,60,82)], [(1,91,55,62),(2,96,56,67),(3,89,57,72),(4,94,58,65),(5,87,59,70),(6,92,60,63),(7,85,49,68),(8,90,50,61),(9,95,51,66),(10,88,52,71),(11,93,53,64),(12,86,54,69),(13,34,48,78),(14,27,37,83),(15,32,38,76),(16,25,39,81),(17,30,40,74),(18,35,41,79),(19,28,42,84),(20,33,43,77),(21,26,44,82),(22,31,45,75),(23,36,46,80),(24,29,47,73)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,61),(13,34),(14,35),(15,36),(16,25),(17,26),(18,27),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 3 | 9 | 0 | 0 | 0 | 0 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 12 | 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 | 11 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [3,9,0,0,0,0,9,10,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,11,1] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | Q8 | D6 | D6 | D6 | D6 | 2- (1+4) | S3×D4 | S3×Q8 | Q8○D12 |
| kernel | Dic6⋊21D4 | Dic3.D4 | Dic3⋊4D4 | Dic6⋊C4 | C12⋊Q8 | D6⋊Q8 | C4.D12 | C4×C3⋊D4 | Dic3⋊Q8 | C3×C22⋊Q8 | C22×Dic6 | C2×S3×Q8 | C22⋊Q8 | Dic6 | C3⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 3 | 1 | 1 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_{21}D_4 % in TeX
G:=Group("Dic6:21D4"); // GroupNames label
G:=SmallGroup(192,1191);
// by ID
G=gap.SmallGroup(192,1191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,570,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^5,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations