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## G = SD16×D11order 352 = 25·11

### Direct product of SD16 and D11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — SD16×D11
 Chief series C1 — C11 — C22 — C44 — C4×D11 — D4×D11 — SD16×D11
 Lower central C11 — C22 — C44 — SD16×D11
 Upper central C1 — C2 — C4 — SD16

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

Smallest permutation representation of 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

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