metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6.16D4, D6⋊C8⋊35C2, C4.65(S3×D4), D6⋊3Q8⋊5C2, (C3×D4).10D4, (C2×C8).149D6, (C2×Q8).79D6, C6.60C22≀C2, (C6×SD16)⋊23C2, (C2×SD16)⋊14S3, (C2×D4).147D6, C6.65(C4○D8), C12.177(C2×D4), C3⋊5(D4.7D4), D4.9(C3⋊D4), D4⋊Dic3⋊35C2, C2.Dic12⋊36C2, (C6×D4).99C22, (C22×S3).38D4, C22.270(S3×D4), (C6×Q8).79C22, C2.28(C23⋊2D6), (C2×C12).450C23, (C2×C24).296C22, (C2×Dic3).185D4, C2.30(D4.D6), C6.50(C8.C22), C2.31(Q8.7D6), C4⋊Dic3.177C22, (C2×Dic6).128C22, C4.45(C2×C3⋊D4), (C2×C3⋊Q16)⋊19C2, (C2×C6).362(C2×D4), (S3×C2×C4).50C22, (C2×D4⋊2S3).6C2, (C2×C3⋊C8).159C22, (C2×C4).539(C22×S3), SmallGroup(192,732)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.16D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=cac-1=a-1, dad=a5, cbc-1=a9b, dbd=a6b, dcd=a6c-1 >
Subgroups: 440 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3⋊Q16, C2×C24, C3×SD16, C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, D4.7D4, C2.Dic12, D6⋊C8, D4⋊Dic3, C2×C3⋊Q16, D6⋊3Q8, C6×SD16, C2×D4⋊2S3, Dic6.16D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C4○D8, C8.C22, S3×D4, C2×C3⋊D4, D4.7D4, D4.D6, Q8.7D6, C23⋊2D6, Dic6.16D4
►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 89 7 95)(2 88 8 94)(3 87 9 93)(4 86 10 92)(5 85 11 91)(6 96 12 90)(13 44 19 38)(14 43 20 37)(15 42 21 48)(16 41 22 47)(17 40 23 46)(18 39 24 45)(25 60 31 54)(26 59 32 53)(27 58 33 52)(28 57 34 51)(29 56 35 50)(30 55 36 49)(61 75 67 81)(62 74 68 80)(63 73 69 79)(64 84 70 78)(65 83 71 77)(66 82 72 76)
(1 64 50 14)(2 63 51 13)(3 62 52 24)(4 61 53 23)(5 72 54 22)(6 71 55 21)(7 70 56 20)(8 69 57 19)(9 68 58 18)(10 67 59 17)(11 66 60 16)(12 65 49 15)(25 38 85 79)(26 37 86 78)(27 48 87 77)(28 47 88 76)(29 46 89 75)(30 45 90 74)(31 44 91 73)(32 43 92 84)(33 42 93 83)(34 41 94 82)(35 40 95 81)(36 39 96 80)
(1 56)(2 49)(3 54)(4 59)(5 52)(6 57)(7 50)(8 55)(9 60)(10 53)(11 58)(12 51)(13 21)(15 19)(16 24)(18 22)(25 93)(26 86)(27 91)(28 96)(29 89)(30 94)(31 87)(32 92)(33 85)(34 90)(35 95)(36 88)(37 43)(38 48)(39 41)(40 46)(42 44)(45 47)(62 66)(63 71)(65 69)(68 72)(73 83)(74 76)(75 81)(77 79)(78 84)(80 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), (1,89,7,95)(2,88,8,94)(3,87,9,93)(4,86,10,92)(5,85,11,91)(6,96,12,90)(13,44,19,38)(14,43,20,37)(15,42,21,48)(16,41,22,47)(17,40,23,46)(18,39,24,45)(25,60,31,54)(26,59,32,53)(27,58,33,52)(28,57,34,51)(29,56,35,50)(30,55,36,49)(61,75,67,81)(62,74,68,80)(63,73,69,79)(64,84,70,78)(65,83,71,77)(66,82,72,76), (1,64,50,14)(2,63,51,13)(3,62,52,24)(4,61,53,23)(5,72,54,22)(6,71,55,21)(7,70,56,20)(8,69,57,19)(9,68,58,18)(10,67,59,17)(11,66,60,16)(12,65,49,15)(25,38,85,79)(26,37,86,78)(27,48,87,77)(28,47,88,76)(29,46,89,75)(30,45,90,74)(31,44,91,73)(32,43,92,84)(33,42,93,83)(34,41,94,82)(35,40,95,81)(36,39,96,80), (1,56)(2,49)(3,54)(4,59)(5,52)(6,57)(7,50)(8,55)(9,60)(10,53)(11,58)(12,51)(13,21)(15,19)(16,24)(18,22)(25,93)(26,86)(27,91)(28,96)(29,89)(30,94)(31,87)(32,92)(33,85)(34,90)(35,95)(36,88)(37,43)(38,48)(39,41)(40,46)(42,44)(45,47)(62,66)(63,71)(65,69)(68,72)(73,83)(74,76)(75,81)(77,79)(78,84)(80,82)>;
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), (1,89,7,95)(2,88,8,94)(3,87,9,93)(4,86,10,92)(5,85,11,91)(6,96,12,90)(13,44,19,38)(14,43,20,37)(15,42,21,48)(16,41,22,47)(17,40,23,46)(18,39,24,45)(25,60,31,54)(26,59,32,53)(27,58,33,52)(28,57,34,51)(29,56,35,50)(30,55,36,49)(61,75,67,81)(62,74,68,80)(63,73,69,79)(64,84,70,78)(65,83,71,77)(66,82,72,76), (1,64,50,14)(2,63,51,13)(3,62,52,24)(4,61,53,23)(5,72,54,22)(6,71,55,21)(7,70,56,20)(8,69,57,19)(9,68,58,18)(10,67,59,17)(11,66,60,16)(12,65,49,15)(25,38,85,79)(26,37,86,78)(27,48,87,77)(28,47,88,76)(29,46,89,75)(30,45,90,74)(31,44,91,73)(32,43,92,84)(33,42,93,83)(34,41,94,82)(35,40,95,81)(36,39,96,80), (1,56)(2,49)(3,54)(4,59)(5,52)(6,57)(7,50)(8,55)(9,60)(10,53)(11,58)(12,51)(13,21)(15,19)(16,24)(18,22)(25,93)(26,86)(27,91)(28,96)(29,89)(30,94)(31,87)(32,92)(33,85)(34,90)(35,95)(36,88)(37,43)(38,48)(39,41)(40,46)(42,44)(45,47)(62,66)(63,71)(65,69)(68,72)(73,83)(74,76)(75,81)(77,79)(78,84)(80,82) );
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)], [(1,89,7,95),(2,88,8,94),(3,87,9,93),(4,86,10,92),(5,85,11,91),(6,96,12,90),(13,44,19,38),(14,43,20,37),(15,42,21,48),(16,41,22,47),(17,40,23,46),(18,39,24,45),(25,60,31,54),(26,59,32,53),(27,58,33,52),(28,57,34,51),(29,56,35,50),(30,55,36,49),(61,75,67,81),(62,74,68,80),(63,73,69,79),(64,84,70,78),(65,83,71,77),(66,82,72,76)], [(1,64,50,14),(2,63,51,13),(3,62,52,24),(4,61,53,23),(5,72,54,22),(6,71,55,21),(7,70,56,20),(8,69,57,19),(9,68,58,18),(10,67,59,17),(11,66,60,16),(12,65,49,15),(25,38,85,79),(26,37,86,78),(27,48,87,77),(28,47,88,76),(29,46,89,75),(30,45,90,74),(31,44,91,73),(32,43,92,84),(33,42,93,83),(34,41,94,82),(35,40,95,81),(36,39,96,80)], [(1,56),(2,49),(3,54),(4,59),(5,52),(6,57),(7,50),(8,55),(9,60),(10,53),(11,58),(12,51),(13,21),(15,19),(16,24),(18,22),(25,93),(26,86),(27,91),(28,96),(29,89),(30,94),(31,87),(32,92),(33,85),(34,90),(35,95),(36,88),(37,43),(38,48),(39,41),(40,46),(42,44),(45,47),(62,66),(63,71),(65,69),(68,72),(73,83),(74,76),(75,81),(77,79),(78,84),(80,82)]])
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 | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 2 | 2 | 2 | 6 | 6 | 8 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C4○D8 | C8.C22 | S3×D4 | S3×D4 | D4.D6 | Q8.7D6 |
| kernel | Dic6.16D4 | C2.Dic12 | D6⋊C8 | D4⋊Dic3 | C2×C3⋊Q16 | D6⋊3Q8 | C6×SD16 | C2×D4⋊2S3 | C2×SD16 | Dic6 | C2×Dic3 | C3×D4 | C22×S3 | C2×C8 | C2×D4 | C2×Q8 | D4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic6.16D4 ►in GL4(𝔽73) generated by
| 46 | 0 | 0 | 0 |
| 4 | 27 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 1 |
| 14 | 43 | 0 | 0 |
| 9 | 59 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
| 57 | 3 | 0 | 0 |
| 61 | 16 | 0 | 0 |
| 0 | 0 | 30 | 60 |
| 0 | 0 | 30 | 43 |
| 72 | 0 | 0 | 0 |
| 38 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(73))| [46,4,0,0,0,27,0,0,0,0,0,1,0,0,72,1],[14,9,0,0,43,59,0,0,0,0,1,72,0,0,0,72],[57,61,0,0,3,16,0,0,0,0,30,30,0,0,60,43],[72,38,0,0,0,1,0,0,0,0,72,1,0,0,0,1] >;
Dic6.16D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6._{16}D_4 % in TeX
G:=Group("Dic6.16D4"); // GroupNames label
G:=SmallGroup(192,732);
// by ID
G=gap.SmallGroup(192,732);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,254,219,184,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^9*b,d*b*d=a^6*b,d*c*d=a^6*c^-1>;
// generators/relations