metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16.6D6, D12.44D4, C24.7C23, SD16.1D6, C12.26C24, Dic6.44D4, D12.19C23, M4(2).18D6, Dic12.1C22, Dic6.19C23, C3⋊4(Q8○D8), Q8○D12⋊8C2, (S3×Q16)⋊2C2, C3⋊D4.7D4, D4⋊S3.C22, D12.C4⋊4C2, C4○D4.31D6, D6.35(C2×D4), Q16⋊S3⋊4C2, C4.118(S3×D4), C8.D6⋊4C2, D4.D6⋊4C2, C8.C22⋊5S3, C8⋊S3.C22, C24⋊C2.C22, C3⋊C8.28C23, C8.7(C22×S3), Q8.7D6⋊4C2, Q8.13D6⋊6C2, C12.247(C2×D4), (S3×C8).2C22, C4.26(S3×C23), (C2×Q8).116D6, C22.17(S3×D4), (C3×SD16).C22, (S3×Q8).3C22, (C4×S3).17C23, Q8.15D6⋊6C2, Dic3.40(C2×D4), (C3×D4).19C23, D4.19(C22×S3), C6.127(C22×D4), D4.S3.2C22, Q8.29(C22×S3), (C3×Q8).19C23, (C3×Q16).1C22, C3⋊Q16.3C22, (C2×C12).117C23, C4○D12.32C22, D4⋊2S3.3C22, (C6×Q8).153C22, Q8⋊2S3.2C22, Q8⋊3S3.3C22, (C3×M4(2)).1C22, (C2×Dic6).201C22, C2.100(C2×S3×D4), (C2×C6).72(C2×D4), (C2×C3⋊Q16)⋊29C2, (C3×C8.C22)⋊4C2, (C2×C3⋊C8).182C22, (C2×C4).101(C22×S3), (C3×C4○D4).28C22, SmallGroup(192,1338)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 608 in 248 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×8], C22, C22 [×4], S3 [×3], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4, D4 [×10], Q8, Q8 [×2], Q8 [×10], Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6, C2×C6, C2×C6, C2×C8 [×3], M4(2), M4(2) [×2], D8, SD16 [×2], SD16 [×4], Q16 [×2], Q16 [×7], C2×Q8, C2×Q8 [×7], C4○D4, C4○D4 [×12], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×2], Dic6 [×5], C4×S3 [×2], C4×S3 [×7], D12 [×2], D12 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×Q8 [×2], C3×Q8, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22, C8.C22 [×5], 2- (1+4) [×2], S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], Dic12 [×2], C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C3⋊Q16 [×4], C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C2×Dic6, C2×Dic6, C4○D12 [×2], C4○D12 [×3], D4⋊2S3 [×2], D4⋊2S3 [×2], S3×Q8 [×4], S3×Q8, Q8⋊3S3 [×2], Q8⋊3S3, C6×Q8, C3×C4○D4, Q8○D8, D12.C4, C8.D6, D4.D6 [×2], Q8.7D6 [×2], S3×Q16 [×2], Q16⋊S3 [×2], C2×C3⋊Q16, Q8.13D6, C3×C8.C22, Q8.15D6, Q8○D12, SD16.D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, S3×D4 [×2], S3×C23, Q8○D8, C2×S3×D4, SD16.D6
Generators and relations
G = < a,b,c,d | a8=b2=c6=1, d2=a4, bab=a3, cac-1=dad-1=a5, cbc-1=a4b, bd=db, dcd-1=a4c-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 61)(2 64)(3 59)(4 62)(5 57)(6 60)(7 63)(8 58)(9 77)(10 80)(11 75)(12 78)(13 73)(14 76)(15 79)(16 74)(17 39)(18 34)(19 37)(20 40)(21 35)(22 38)(23 33)(24 36)(25 47)(26 42)(27 45)(28 48)(29 43)(30 46)(31 41)(32 44)(49 94)(50 89)(51 92)(52 95)(53 90)(54 93)(55 96)(56 91)(65 83)(66 86)(67 81)(68 84)(69 87)(70 82)(71 85)(72 88)
(1 85 49 33 73 42)(2 82 50 38 74 47)(3 87 51 35 75 44)(4 84 52 40 76 41)(5 81 53 37 77 46)(6 86 54 34 78 43)(7 83 55 39 79 48)(8 88 56 36 80 45)(9 26 57 71 90 23)(10 31 58 68 91 20)(11 28 59 65 92 17)(12 25 60 70 93 22)(13 30 61 67 94 19)(14 27 62 72 95 24)(15 32 63 69 96 21)(16 29 64 66 89 18)
(1 25 5 29)(2 30 6 26)(3 27 7 31)(4 32 8 28)(9 86 13 82)(10 83 14 87)(11 88 15 84)(12 85 16 81)(17 52 21 56)(18 49 22 53)(19 54 23 50)(20 51 24 55)(33 89 37 93)(34 94 38 90)(35 91 39 95)(36 96 40 92)(41 59 45 63)(42 64 46 60)(43 61 47 57)(44 58 48 62)(65 76 69 80)(66 73 70 77)(67 78 71 74)(68 75 72 79)
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), (1,61)(2,64)(3,59)(4,62)(5,57)(6,60)(7,63)(8,58)(9,77)(10,80)(11,75)(12,78)(13,73)(14,76)(15,79)(16,74)(17,39)(18,34)(19,37)(20,40)(21,35)(22,38)(23,33)(24,36)(25,47)(26,42)(27,45)(28,48)(29,43)(30,46)(31,41)(32,44)(49,94)(50,89)(51,92)(52,95)(53,90)(54,93)(55,96)(56,91)(65,83)(66,86)(67,81)(68,84)(69,87)(70,82)(71,85)(72,88), (1,85,49,33,73,42)(2,82,50,38,74,47)(3,87,51,35,75,44)(4,84,52,40,76,41)(5,81,53,37,77,46)(6,86,54,34,78,43)(7,83,55,39,79,48)(8,88,56,36,80,45)(9,26,57,71,90,23)(10,31,58,68,91,20)(11,28,59,65,92,17)(12,25,60,70,93,22)(13,30,61,67,94,19)(14,27,62,72,95,24)(15,32,63,69,96,21)(16,29,64,66,89,18), (1,25,5,29)(2,30,6,26)(3,27,7,31)(4,32,8,28)(9,86,13,82)(10,83,14,87)(11,88,15,84)(12,85,16,81)(17,52,21,56)(18,49,22,53)(19,54,23,50)(20,51,24,55)(33,89,37,93)(34,94,38,90)(35,91,39,95)(36,96,40,92)(41,59,45,63)(42,64,46,60)(43,61,47,57)(44,58,48,62)(65,76,69,80)(66,73,70,77)(67,78,71,74)(68,75,72,79)>;
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), (1,61)(2,64)(3,59)(4,62)(5,57)(6,60)(7,63)(8,58)(9,77)(10,80)(11,75)(12,78)(13,73)(14,76)(15,79)(16,74)(17,39)(18,34)(19,37)(20,40)(21,35)(22,38)(23,33)(24,36)(25,47)(26,42)(27,45)(28,48)(29,43)(30,46)(31,41)(32,44)(49,94)(50,89)(51,92)(52,95)(53,90)(54,93)(55,96)(56,91)(65,83)(66,86)(67,81)(68,84)(69,87)(70,82)(71,85)(72,88), (1,85,49,33,73,42)(2,82,50,38,74,47)(3,87,51,35,75,44)(4,84,52,40,76,41)(5,81,53,37,77,46)(6,86,54,34,78,43)(7,83,55,39,79,48)(8,88,56,36,80,45)(9,26,57,71,90,23)(10,31,58,68,91,20)(11,28,59,65,92,17)(12,25,60,70,93,22)(13,30,61,67,94,19)(14,27,62,72,95,24)(15,32,63,69,96,21)(16,29,64,66,89,18), (1,25,5,29)(2,30,6,26)(3,27,7,31)(4,32,8,28)(9,86,13,82)(10,83,14,87)(11,88,15,84)(12,85,16,81)(17,52,21,56)(18,49,22,53)(19,54,23,50)(20,51,24,55)(33,89,37,93)(34,94,38,90)(35,91,39,95)(36,96,40,92)(41,59,45,63)(42,64,46,60)(43,61,47,57)(44,58,48,62)(65,76,69,80)(66,73,70,77)(67,78,71,74)(68,75,72,79) );
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)], [(1,61),(2,64),(3,59),(4,62),(5,57),(6,60),(7,63),(8,58),(9,77),(10,80),(11,75),(12,78),(13,73),(14,76),(15,79),(16,74),(17,39),(18,34),(19,37),(20,40),(21,35),(22,38),(23,33),(24,36),(25,47),(26,42),(27,45),(28,48),(29,43),(30,46),(31,41),(32,44),(49,94),(50,89),(51,92),(52,95),(53,90),(54,93),(55,96),(56,91),(65,83),(66,86),(67,81),(68,84),(69,87),(70,82),(71,85),(72,88)], [(1,85,49,33,73,42),(2,82,50,38,74,47),(3,87,51,35,75,44),(4,84,52,40,76,41),(5,81,53,37,77,46),(6,86,54,34,78,43),(7,83,55,39,79,48),(8,88,56,36,80,45),(9,26,57,71,90,23),(10,31,58,68,91,20),(11,28,59,65,92,17),(12,25,60,70,93,22),(13,30,61,67,94,19),(14,27,62,72,95,24),(15,32,63,69,96,21),(16,29,64,66,89,18)], [(1,25,5,29),(2,30,6,26),(3,27,7,31),(4,32,8,28),(9,86,13,82),(10,83,14,87),(11,88,15,84),(12,85,16,81),(17,52,21,56),(18,49,22,53),(19,54,23,50),(20,51,24,55),(33,89,37,93),(34,94,38,90),(35,91,39,95),(36,96,40,92),(41,59,45,63),(42,64,46,60),(43,61,47,57),(44,58,48,62),(65,76,69,80),(66,73,70,77),(67,78,71,74),(68,75,72,79)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 71 | 0 | 34 |
| 0 | 0 | 1 | 0 | 56 | 17 |
| 0 | 0 | 55 | 1 | 72 | 57 |
| 0 | 0 | 71 | 57 | 72 | 55 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 32 | 0 | 0 |
| 0 | 0 | 57 | 32 | 0 | 0 |
| 0 | 0 | 32 | 57 | 16 | 16 |
| 0 | 0 | 16 | 57 | 16 | 57 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 32 | 32 |
| 0 | 0 | 16 | 0 | 0 | 32 |
| 0 | 0 | 0 | 57 | 57 | 57 |
| 0 | 0 | 57 | 16 | 57 | 57 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 2 | 71 |
| 0 | 0 | 1 | 17 | 2 | 0 |
| 0 | 0 | 56 | 1 | 55 | 1 |
| 0 | 0 | 55 | 1 | 72 | 57 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,19,1,55,71,0,0,71,0,1,57,0,0,0,56,72,72,0,0,34,17,57,55],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,41,57,32,16,0,0,32,32,57,57,0,0,0,0,16,16,0,0,0,0,16,57],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,32,16,0,57,0,0,0,0,57,16,0,0,32,0,57,57,0,0,32,32,57,57],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,17,1,56,55,0,0,0,17,1,1,0,0,2,2,55,72,0,0,71,0,1,57] >;
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 4 | 8 | 4 | 4 | 6 | 6 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | S3×D4 | S3×D4 | Q8○D8 | SD16.D6 |
| kernel | SD16.D6 | D12.C4 | C8.D6 | D4.D6 | Q8.7D6 | S3×Q16 | Q16⋊S3 | C2×C3⋊Q16 | Q8.13D6 | C3×C8.C22 | Q8.15D6 | Q8○D12 | C8.C22 | Dic6 | D12 | C3⋊D4 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 |
In GAP, Magma, Sage, TeX
SD_{16}.D_6 % in TeX
G:=Group("SD16.D6"); // GroupNames label
G:=SmallGroup(192,1338);
// by ID
G=gap.SmallGroup(192,1338);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,184,570,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^4,b*a*b=a^3,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations