metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6Dic6, C42.107D6, C6.1012+ (1+4), (C3×D4)⋊7Q8, C4⋊C4.281D6, C3⋊3(D4⋊3Q8), (C4×D4).14S3, C12.43(C2×Q8), C12⋊2Q8⋊23C2, (C4×Dic6)⋊29C2, (C2×D4).243D6, (D4×C12).15C2, (C2×C6).87C24, C4.16(C2×Dic6), C2.13(D4○D12), C12.48D4⋊9C2, C6.14(C22×Q8), C22⋊C4.108D6, C12.3Q8⋊15C2, (D4×Dic3).12C2, (C22×C4).222D6, C12.292(C4○D4), C22.2(C2×Dic6), (C4×C12).149C22, (C2×C12).156C23, Dic3.D4⋊8C2, (C6×D4).251C22, Dic3⋊C4.6C22, C4.117(D4⋊2S3), C2.16(C22×Dic6), C4⋊Dic3.198C22, C22.115(S3×C23), (C22×C12).80C22, C23.176(C22×S3), (C22×C6).157C23, (C2×Dic6).26C22, (C4×Dic3).74C22, (C2×Dic3).37C23, C6.D4.10C22, (C22×Dic3).94C22, (C2×C6).4(C2×Q8), C6.73(C2×C4○D4), (C2×C4⋊Dic3)⋊24C2, C2.21(C2×D4⋊2S3), (C3×C4⋊C4).323C22, (C2×C4).731(C22×S3), (C3×C22⋊C4).105C22, SmallGroup(192,1102)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 504 in 228 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×11], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×8], C12 [×4], C12 [×3], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×3], Dic6 [×4], C2×Dic3 [×8], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×C6 [×2], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×8], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C2×Dic6 [×2], C22×Dic3 [×4], C22×C12 [×2], C6×D4, D4⋊3Q8, C4×Dic6, C12⋊2Q8, Dic3.D4 [×4], C12.3Q8 [×2], C12.48D4 [×2], C2×C4⋊Dic3 [×2], D4×Dic3 [×2], D4×C12, D4⋊6Dic6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C2×Dic6 [×6], D4⋊2S3 [×2], S3×C23, D4⋊3Q8, C22×Dic6, C2×D4⋊2S3, D4○D12, D4⋊6Dic6
Generators and relations
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
►On 96 points
(1 27 52 67)(2 28 53 68)(3 29 54 69)(4 30 55 70)(5 31 56 71)(6 32 57 72)(7 33 58 61)(8 34 59 62)(9 35 60 63)(10 36 49 64)(11 25 50 65)(12 26 51 66)(13 78 88 44)(14 79 89 45)(15 80 90 46)(16 81 91 47)(17 82 92 48)(18 83 93 37)(19 84 94 38)(20 73 95 39)(21 74 96 40)(22 75 85 41)(23 76 86 42)(24 77 87 43)
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 85)(48 86)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(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 80 7 74)(2 79 8 73)(3 78 9 84)(4 77 10 83)(5 76 11 82)(6 75 12 81)(13 63 19 69)(14 62 20 68)(15 61 21 67)(16 72 22 66)(17 71 23 65)(18 70 24 64)(25 92 31 86)(26 91 32 85)(27 90 33 96)(28 89 34 95)(29 88 35 94)(30 87 36 93)(37 55 43 49)(38 54 44 60)(39 53 45 59)(40 52 46 58)(41 51 47 57)(42 50 48 56)
G:=sub<Sym(96)| (1,27,52,67)(2,28,53,68)(3,29,54,69)(4,30,55,70)(5,31,56,71)(6,32,57,72)(7,33,58,61)(8,34,59,62)(9,35,60,63)(10,36,49,64)(11,25,50,65)(12,26,51,66)(13,78,88,44)(14,79,89,45)(15,80,90,46)(16,81,91,47)(17,82,92,48)(18,83,93,37)(19,84,94,38)(20,73,95,39)(21,74,96,40)(22,75,85,41)(23,76,86,42)(24,77,87,43), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,85)(48,86)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (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), (1,80,7,74)(2,79,8,73)(3,78,9,84)(4,77,10,83)(5,76,11,82)(6,75,12,81)(13,63,19,69)(14,62,20,68)(15,61,21,67)(16,72,22,66)(17,71,23,65)(18,70,24,64)(25,92,31,86)(26,91,32,85)(27,90,33,96)(28,89,34,95)(29,88,35,94)(30,87,36,93)(37,55,43,49)(38,54,44,60)(39,53,45,59)(40,52,46,58)(41,51,47,57)(42,50,48,56)>;
G:=Group( (1,27,52,67)(2,28,53,68)(3,29,54,69)(4,30,55,70)(5,31,56,71)(6,32,57,72)(7,33,58,61)(8,34,59,62)(9,35,60,63)(10,36,49,64)(11,25,50,65)(12,26,51,66)(13,78,88,44)(14,79,89,45)(15,80,90,46)(16,81,91,47)(17,82,92,48)(18,83,93,37)(19,84,94,38)(20,73,95,39)(21,74,96,40)(22,75,85,41)(23,76,86,42)(24,77,87,43), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,85)(48,86)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (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), (1,80,7,74)(2,79,8,73)(3,78,9,84)(4,77,10,83)(5,76,11,82)(6,75,12,81)(13,63,19,69)(14,62,20,68)(15,61,21,67)(16,72,22,66)(17,71,23,65)(18,70,24,64)(25,92,31,86)(26,91,32,85)(27,90,33,96)(28,89,34,95)(29,88,35,94)(30,87,36,93)(37,55,43,49)(38,54,44,60)(39,53,45,59)(40,52,46,58)(41,51,47,57)(42,50,48,56) );
G=PermutationGroup([(1,27,52,67),(2,28,53,68),(3,29,54,69),(4,30,55,70),(5,31,56,71),(6,32,57,72),(7,33,58,61),(8,34,59,62),(9,35,60,63),(10,36,49,64),(11,25,50,65),(12,26,51,66),(13,78,88,44),(14,79,89,45),(15,80,90,46),(16,81,91,47),(17,82,92,48),(18,83,93,37),(19,84,94,38),(20,73,95,39),(21,74,96,40),(22,75,85,41),(23,76,86,42),(24,77,87,43)], [(1,33),(2,34),(3,35),(4,36),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,84),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,85),(48,86),(49,70),(50,71),(51,72),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69)], [(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)], [(1,80,7,74),(2,79,8,73),(3,78,9,84),(4,77,10,83),(5,76,11,82),(6,75,12,81),(13,63,19,69),(14,62,20,68),(15,61,21,67),(16,72,22,66),(17,71,23,65),(18,70,24,64),(25,92,31,86),(26,91,32,85),(27,90,33,96),(28,89,34,95),(29,88,35,94),(30,87,36,93),(37,55,43,49),(38,54,44,60),(39,53,45,59),(40,52,46,58),(41,51,47,57),(42,50,48,56)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 |
| 7 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 11 | 0 | 0 |
| 12 | 7 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,0,12,0,0,12,0],[7,2,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[6,12,0,0,11,7,0,0,0,0,0,8,0,0,5,0] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | D6 | D6 | C4○D4 | Dic6 | 2+ (1+4) | D4⋊2S3 | D4○D12 |
| kernel | D4⋊6Dic6 | C4×Dic6 | C12⋊2Q8 | Dic3.D4 | C12.3Q8 | C12.48D4 | C2×C4⋊Dic3 | D4×Dic3 | D4×C12 | C4×D4 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | D4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_4\rtimes_6Dic_6
% in TeX
G:=Group("D4:6Dic6"); // GroupNames label
G:=SmallGroup(192,1102);
// by ID
G=gap.SmallGroup(192,1102);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,1571,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations