direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16×D11, C8⋊5D22, Q8⋊1D22, C88⋊5C22, D4.2D22, D22.13D4, C44.4C23, Dic11.4D4, D44.2C22, Dic22⋊2C22, (D4×D11).C2, (C8×D11)⋊4C2, C11⋊C8⋊6C22, Q8⋊D11⋊1C2, (Q8×D11)⋊1C2, C8⋊D11⋊5C2, C11⋊2(C2×SD16), D4.D11⋊3C2, C22.30(C2×D4), C2.18(D4×D11), (C11×SD16)⋊3C2, (Q8×C11)⋊1C22, C4.4(C22×D11), (C4×D11).9C22, (D4×C11).2C22, SmallGroup(352,108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×D11
G = < a,b,c,d | a8=b2=c11=d2=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 498 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C11, C2×C8, SD16, SD16, C2×D4, C2×Q8, D11, D11, C22, C22, C2×SD16, Dic11, Dic11, C44, C44, D22, D22, C2×C22, C11⋊C8, C88, Dic22, Dic22, C4×D11, C4×D11, D44, C11⋊D4, D4×C11, Q8×C11, C22×D11, C8×D11, C8⋊D11, D4.D11, Q8⋊D11, C11×SD16, D4×D11, Q8×D11, SD16×D11
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, D11, C2×SD16, D22, C22×D11, D4×D11, SD16×D11
►On 88 points
Generators in S88
(1 87 32 65 21 76 43 54)(2 88 33 66 22 77 44 55)(3 78 23 56 12 67 34 45)(4 79 24 57 13 68 35 46)(5 80 25 58 14 69 36 47)(6 81 26 59 15 70 37 48)(7 82 27 60 16 71 38 49)(8 83 28 61 17 72 39 50)(9 84 29 62 18 73 40 51)(10 85 30 63 19 74 41 52)(11 86 31 64 20 75 42 53)
(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)
G:=sub<Sym(88)| (1,87,32,65,21,76,43,54)(2,88,33,66,22,77,44,55)(3,78,23,56,12,67,34,45)(4,79,24,57,13,68,35,46)(5,80,25,58,14,69,36,47)(6,81,26,59,15,70,37,48)(7,82,27,60,16,71,38,49)(8,83,28,61,17,72,39,50)(9,84,29,62,18,73,40,51)(10,85,30,63,19,74,41,52)(11,86,31,64,20,75,42,53), (23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)>;
G:=Group( (1,87,32,65,21,76,43,54)(2,88,33,66,22,77,44,55)(3,78,23,56,12,67,34,45)(4,79,24,57,13,68,35,46)(5,80,25,58,14,69,36,47)(6,81,26,59,15,70,37,48)(7,82,27,60,16,71,38,49)(8,83,28,61,17,72,39,50)(9,84,29,62,18,73,40,51)(10,85,30,63,19,74,41,52)(11,86,31,64,20,75,42,53), (23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85) );
G=PermutationGroup([[(1,87,32,65,21,76,43,54),(2,88,33,66,22,77,44,55),(3,78,23,56,12,67,34,45),(4,79,24,57,13,68,35,46),(5,80,25,58,14,69,36,47),(6,81,26,59,15,70,37,48),(7,82,27,60,16,71,38,49),(8,83,28,61,17,72,39,50),(9,84,29,62,18,73,40,51),(10,85,30,63,19,74,41,52),(11,86,31,64,20,75,42,53)], [(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,22),(11,21),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,44),(31,43),(32,42),(33,41),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,66),(53,65),(54,64),(55,63),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,88),(75,87),(76,86),(77,85)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22J | 44A | ··· | 44E | 44F | ··· | 44J | 88A | ··· | 88J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 4 | 11 | 11 | 44 | 2 | 4 | 22 | 44 | 2 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | D11 | D22 | D22 | D22 | D4×D11 | SD16×D11 |
| kernel | SD16×D11 | C8×D11 | C8⋊D11 | D4.D11 | Q8⋊D11 | C11×SD16 | D4×D11 | Q8×D11 | Dic11 | D22 | D11 | SD16 | C8 | D4 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 5 | 5 | 5 | 5 | 5 | 10 |
Matrix representation of SD16×D11 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 83 | 76 |
| 0 | 0 | 39 | 55 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 86 |
| 0 | 0 | 0 | 88 |
| 3 | 1 | 0 | 0 |
| 11 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 88 | 0 | 0 |
| 15 | 85 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,83,39,0,0,76,55],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,86,88],[3,11,0,0,1,4,0,0,0,0,1,0,0,0,0,1],[4,15,0,0,88,85,0,0,0,0,1,0,0,0,0,1] >;
SD16×D11 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times D_{11} % in TeX
G:=Group("SD16xD11"); // GroupNames label
G:=SmallGroup(352,108);
// by ID
G=gap.SmallGroup(352,108);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,116,86,297,159,69,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^11=d^2=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations