Copied to
clipboard

## G = Dic3⋊8SD16order 192 = 26·3

### 3rd semidirect product of Dic3 and SD16 acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Dic3⋊8SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — Dic3⋊5D4 — Dic3⋊8SD16
 Lower central C3 — C6 — C12 — Dic3⋊8SD16
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for Dic38SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c3 >

Subgroups: 352 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C24⋊C2, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4×SD16, C6.D8, C6.SD16, C8×Dic3, C3×C4.Q8, Dic6⋊C4, Dic35D4, C2×C24⋊C2, Dic38SD16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, SD16, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C2×SD16, C4○D8, S3×C2×C4, S3×D4, Q83S3, C4×SD16, Dic35D4, S3×SD16, Q8.7D6, Dic38SD16

Smallest permutation representation of Dic38SD16
On 96 points
Generators in S96
(1 73 55 86 34 16)(2 74 56 87 35 9)(3 75 49 88 36 10)(4 76 50 81 37 11)(5 77 51 82 38 12)(6 78 52 83 39 13)(7 79 53 84 40 14)(8 80 54 85 33 15)(17 92 59 47 27 70)(18 93 60 48 28 71)(19 94 61 41 29 72)(20 95 62 42 30 65)(21 96 63 43 31 66)(22 89 64 44 32 67)(23 90 57 45 25 68)(24 91 58 46 26 69)
(1 45 86 23)(2 46 87 24)(3 47 88 17)(4 48 81 18)(5 41 82 19)(6 42 83 20)(7 43 84 21)(8 44 85 22)(9 26 56 91)(10 27 49 92)(11 28 50 93)(12 29 51 94)(13 30 52 95)(14 31 53 96)(15 32 54 89)(16 25 55 90)(33 67 80 64)(34 68 73 57)(35 69 74 58)(36 70 75 59)(37 71 76 60)(38 72 77 61)(39 65 78 62)(40 66 79 63)
(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 82)(2 85)(3 88)(4 83)(5 86)(6 81)(7 84)(8 87)(9 33)(10 36)(11 39)(12 34)(13 37)(14 40)(15 35)(16 38)(17 47)(18 42)(19 45)(20 48)(21 43)(22 46)(23 41)(24 44)(25 72)(26 67)(27 70)(28 65)(29 68)(30 71)(31 66)(32 69)(49 75)(50 78)(51 73)(52 76)(53 79)(54 74)(55 77)(56 80)(57 94)(58 89)(59 92)(60 95)(61 90)(62 93)(63 96)(64 91)

G:=sub<Sym(96)| (1,73,55,86,34,16)(2,74,56,87,35,9)(3,75,49,88,36,10)(4,76,50,81,37,11)(5,77,51,82,38,12)(6,78,52,83,39,13)(7,79,53,84,40,14)(8,80,54,85,33,15)(17,92,59,47,27,70)(18,93,60,48,28,71)(19,94,61,41,29,72)(20,95,62,42,30,65)(21,96,63,43,31,66)(22,89,64,44,32,67)(23,90,57,45,25,68)(24,91,58,46,26,69), (1,45,86,23)(2,46,87,24)(3,47,88,17)(4,48,81,18)(5,41,82,19)(6,42,83,20)(7,43,84,21)(8,44,85,22)(9,26,56,91)(10,27,49,92)(11,28,50,93)(12,29,51,94)(13,30,52,95)(14,31,53,96)(15,32,54,89)(16,25,55,90)(33,67,80,64)(34,68,73,57)(35,69,74,58)(36,70,75,59)(37,71,76,60)(38,72,77,61)(39,65,78,62)(40,66,79,63), (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), (1,82)(2,85)(3,88)(4,83)(5,86)(6,81)(7,84)(8,87)(9,33)(10,36)(11,39)(12,34)(13,37)(14,40)(15,35)(16,38)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,72)(26,67)(27,70)(28,65)(29,68)(30,71)(31,66)(32,69)(49,75)(50,78)(51,73)(52,76)(53,79)(54,74)(55,77)(56,80)(57,94)(58,89)(59,92)(60,95)(61,90)(62,93)(63,96)(64,91)>;

G:=Group( (1,73,55,86,34,16)(2,74,56,87,35,9)(3,75,49,88,36,10)(4,76,50,81,37,11)(5,77,51,82,38,12)(6,78,52,83,39,13)(7,79,53,84,40,14)(8,80,54,85,33,15)(17,92,59,47,27,70)(18,93,60,48,28,71)(19,94,61,41,29,72)(20,95,62,42,30,65)(21,96,63,43,31,66)(22,89,64,44,32,67)(23,90,57,45,25,68)(24,91,58,46,26,69), (1,45,86,23)(2,46,87,24)(3,47,88,17)(4,48,81,18)(5,41,82,19)(6,42,83,20)(7,43,84,21)(8,44,85,22)(9,26,56,91)(10,27,49,92)(11,28,50,93)(12,29,51,94)(13,30,52,95)(14,31,53,96)(15,32,54,89)(16,25,55,90)(33,67,80,64)(34,68,73,57)(35,69,74,58)(36,70,75,59)(37,71,76,60)(38,72,77,61)(39,65,78,62)(40,66,79,63), (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), (1,82)(2,85)(3,88)(4,83)(5,86)(6,81)(7,84)(8,87)(9,33)(10,36)(11,39)(12,34)(13,37)(14,40)(15,35)(16,38)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,72)(26,67)(27,70)(28,65)(29,68)(30,71)(31,66)(32,69)(49,75)(50,78)(51,73)(52,76)(53,79)(54,74)(55,77)(56,80)(57,94)(58,89)(59,92)(60,95)(61,90)(62,93)(63,96)(64,91) );

G=PermutationGroup([[(1,73,55,86,34,16),(2,74,56,87,35,9),(3,75,49,88,36,10),(4,76,50,81,37,11),(5,77,51,82,38,12),(6,78,52,83,39,13),(7,79,53,84,40,14),(8,80,54,85,33,15),(17,92,59,47,27,70),(18,93,60,48,28,71),(19,94,61,41,29,72),(20,95,62,42,30,65),(21,96,63,43,31,66),(22,89,64,44,32,67),(23,90,57,45,25,68),(24,91,58,46,26,69)], [(1,45,86,23),(2,46,87,24),(3,47,88,17),(4,48,81,18),(5,41,82,19),(6,42,83,20),(7,43,84,21),(8,44,85,22),(9,26,56,91),(10,27,49,92),(11,28,50,93),(12,29,51,94),(13,30,52,95),(14,31,53,96),(15,32,54,89),(16,25,55,90),(33,67,80,64),(34,68,73,57),(35,69,74,58),(36,70,75,59),(37,71,76,60),(38,72,77,61),(39,65,78,62),(40,66,79,63)], [(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)], [(1,82),(2,85),(3,88),(4,83),(5,86),(6,81),(7,84),(8,87),(9,33),(10,36),(11,39),(12,34),(13,37),(14,40),(15,35),(16,38),(17,47),(18,42),(19,45),(20,48),(21,43),(22,46),(23,41),(24,44),(25,72),(26,67),(27,70),(28,65),(29,68),(30,71),(31,66),(32,69),(49,75),(50,78),(51,73),(52,76),(53,79),(54,74),(55,77),(56,80),(57,94),(58,89),(59,92),(60,95),(61,90),(62,93),(63,96),(64,91)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 1 1 12 12 2 2 2 3 3 3 3 4 4 4 4 6 6 12 12 2 2 2 2 2 2 2 6 6 6 6 4 4 8 8 8 8 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 SD16 C4○D4 C4×S3 C4○D8 Q8⋊3S3 S3×D4 S3×SD16 Q8.7D6 kernel Dic3⋊8SD16 C6.D8 C6.SD16 C8×Dic3 C3×C4.Q8 Dic6⋊C4 Dic3⋊5D4 C2×C24⋊C2 C24⋊C2 C4.Q8 C2×Dic3 C4⋊C4 C2×C8 Dic3 C12 C8 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 2 1 4 2 4 4 1 1 2 2

Matrix representation of Dic38SD16 in GL6(𝔽73)

 72 0 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 0 1 0 0 0 0 72 72
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72
,
 40 35 0 0 0 0 21 33 0 0 0 0 0 0 61 4 0 0 0 0 55 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 0 0 0 0 0 44 1 0 0 0 0 0 0 72 0 0 0 0 0 70 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[40,21,0,0,0,0,35,33,0,0,0,0,0,0,61,55,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,44,0,0,0,0,0,1,0,0,0,0,0,0,72,70,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

Dic38SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("Dic3:8SD16");
// GroupNames label

G:=SmallGroup(192,411);
// by ID

G=gap.SmallGroup(192,411);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,120,135,268,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

׿
×
𝔽