metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊17D4, C4⋊C4.60D6, (C2×C6)⋊3SD16, (C2×D4).40D6, C3⋊3(Q8⋊D4), C4.100(S3×D4), (C2×C12).73D4, C4⋊D4.5S3, C6.46C22≀C2, C12.149(C2×D4), C6.55(C2×SD16), (C22×C6).86D4, C6.SD16⋊34C2, C22⋊3(D4.S3), (C6×D4).56C22, (C22×C4).138D6, C12.55D4⋊12C2, C2.14(C23⋊2D6), (C2×C12).359C23, (C22×Dic6)⋊13C2, C23.66(C3⋊D4), C2.12(Q8.14D6), C6.114(C8.C22), (C22×C12).163C22, (C2×Dic6).269C22, (C2×D4.S3)⋊9C2, C2.9(C2×D4.S3), (C3×C4⋊D4).4C2, (C2×C6).490(C2×D4), (C2×C4).51(C3⋊D4), (C2×C3⋊C8).110C22, (C3×C4⋊C4).107C22, (C2×C4).459(C22×S3), C22.165(C2×C3⋊D4), SmallGroup(192,599)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊17D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, bd=db, dcd=c-1 >
Subgroups: 432 in 158 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C2×C3⋊C8, D4.S3, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×D4, C6×D4, Q8⋊D4, C6.SD16, C12.55D4, C2×D4.S3, C3×C4⋊D4, C22×Dic6, Dic6⋊17D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×SD16, C8.C22, D4.S3, S3×D4, C2×C3⋊D4, Q8⋊D4, C2×D4.S3, C23⋊2D6, Q8.14D6, Dic6⋊17D4
►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 35 7 29)(2 34 8 28)(3 33 9 27)(4 32 10 26)(5 31 11 25)(6 30 12 36)(13 68 19 62)(14 67 20 61)(15 66 21 72)(16 65 22 71)(17 64 23 70)(18 63 24 69)(37 77 43 83)(38 76 44 82)(39 75 45 81)(40 74 46 80)(41 73 47 79)(42 84 48 78)(49 94 55 88)(50 93 56 87)(51 92 57 86)(52 91 58 85)(53 90 59 96)(54 89 60 95)
(1 91 46 16)(2 86 47 23)(3 93 48 18)(4 88 37 13)(5 95 38 20)(6 90 39 15)(7 85 40 22)(8 92 41 17)(9 87 42 24)(10 94 43 19)(11 89 44 14)(12 96 45 21)(25 51 82 70)(26 58 83 65)(27 53 84 72)(28 60 73 67)(29 55 74 62)(30 50 75 69)(31 57 76 64)(32 52 77 71)(33 59 78 66)(34 54 79 61)(35 49 80 68)(36 56 81 63)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 94)(38 95)(39 96)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 83)(50 84)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)
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), (1,35,7,29)(2,34,8,28)(3,33,9,27)(4,32,10,26)(5,31,11,25)(6,30,12,36)(13,68,19,62)(14,67,20,61)(15,66,21,72)(16,65,22,71)(17,64,23,70)(18,63,24,69)(37,77,43,83)(38,76,44,82)(39,75,45,81)(40,74,46,80)(41,73,47,79)(42,84,48,78)(49,94,55,88)(50,93,56,87)(51,92,57,86)(52,91,58,85)(53,90,59,96)(54,89,60,95), (1,91,46,16)(2,86,47,23)(3,93,48,18)(4,88,37,13)(5,95,38,20)(6,90,39,15)(7,85,40,22)(8,92,41,17)(9,87,42,24)(10,94,43,19)(11,89,44,14)(12,96,45,21)(25,51,82,70)(26,58,83,65)(27,53,84,72)(28,60,73,67)(29,55,74,62)(30,50,75,69)(31,57,76,64)(32,52,77,71)(33,59,78,66)(34,54,79,61)(35,49,80,68)(36,56,81,63), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,94)(38,95)(39,96)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,83)(50,84)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)>;
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), (1,35,7,29)(2,34,8,28)(3,33,9,27)(4,32,10,26)(5,31,11,25)(6,30,12,36)(13,68,19,62)(14,67,20,61)(15,66,21,72)(16,65,22,71)(17,64,23,70)(18,63,24,69)(37,77,43,83)(38,76,44,82)(39,75,45,81)(40,74,46,80)(41,73,47,79)(42,84,48,78)(49,94,55,88)(50,93,56,87)(51,92,57,86)(52,91,58,85)(53,90,59,96)(54,89,60,95), (1,91,46,16)(2,86,47,23)(3,93,48,18)(4,88,37,13)(5,95,38,20)(6,90,39,15)(7,85,40,22)(8,92,41,17)(9,87,42,24)(10,94,43,19)(11,89,44,14)(12,96,45,21)(25,51,82,70)(26,58,83,65)(27,53,84,72)(28,60,73,67)(29,55,74,62)(30,50,75,69)(31,57,76,64)(32,52,77,71)(33,59,78,66)(34,54,79,61)(35,49,80,68)(36,56,81,63), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,94)(38,95)(39,96)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,83)(50,84)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82) );
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)], [(1,35,7,29),(2,34,8,28),(3,33,9,27),(4,32,10,26),(5,31,11,25),(6,30,12,36),(13,68,19,62),(14,67,20,61),(15,66,21,72),(16,65,22,71),(17,64,23,70),(18,63,24,69),(37,77,43,83),(38,76,44,82),(39,75,45,81),(40,74,46,80),(41,73,47,79),(42,84,48,78),(49,94,55,88),(50,93,56,87),(51,92,57,86),(52,91,58,85),(53,90,59,96),(54,89,60,95)], [(1,91,46,16),(2,86,47,23),(3,93,48,18),(4,88,37,13),(5,95,38,20),(6,90,39,15),(7,85,40,22),(8,92,41,17),(9,87,42,24),(10,94,43,19),(11,89,44,14),(12,96,45,21),(25,51,82,70),(26,58,83,65),(27,53,84,72),(28,60,73,67),(29,55,74,62),(30,50,75,69),(31,57,76,64),(32,52,77,71),(33,59,78,66),(34,54,79,61),(35,49,80,68),(36,56,81,63)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,94),(38,95),(39,96),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,83),(50,84),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | SD16 | C3⋊D4 | C3⋊D4 | C8.C22 | S3×D4 | D4.S3 | Q8.14D6 |
| kernel | Dic6⋊17D4 | C6.SD16 | C12.55D4 | C2×D4.S3 | C3×C4⋊D4 | C22×Dic6 | C4⋊D4 | Dic6 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of Dic6⋊17D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 67 |
| 0 | 0 | 0 | 0 | 67 | 67 |
| 1 | 71 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 71 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,65,0,0,0,0,0,0,9,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,67,0,0,0,0,67,67],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1],[1,0,0,0,0,0,71,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
Dic6⋊17D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_{17}D_4 % in TeX
G:=Group("Dic6:17D4"); // GroupNames label
G:=SmallGroup(192,599);
// by ID
G=gap.SmallGroup(192,599);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,1123,297,136,6278]);
// 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^7,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations