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## G = D6⋊SD16order 192 = 26·3

### 1st semidirect product of D6 and SD16 acting via SD16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D6⋊SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C2×C8 — D6⋊SD16
 Lower central C3 — C6 — C2×C12 — D6⋊SD16
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D6⋊SD16
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a4b, dbd=a3b, dcd=c3 >

Subgroups: 376 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, S3×C8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D4.S3, C6.D4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C6×D4, C88D4, C12.Q8, C2.Dic12, C3×D4⋊C4, C4.D12, S3×C2×C8, C2×D4.S3, D63D4, D6⋊SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C22×S3, C4⋊D4, C2×SD16, C4○D8, C4○D12, S3×D4, C88D4, Dic3⋊D4, D83S3, S3×SD16, D6⋊SD16

Smallest permutation representation of D6⋊SD16
On 96 points
Generators in S96
(1 10 78 53 26 85)(2 86 27 54 79 11)(3 12 80 55 28 87)(4 88 29 56 73 13)(5 14 74 49 30 81)(6 82 31 50 75 15)(7 16 76 51 32 83)(8 84 25 52 77 9)(17 65 39 41 64 95)(18 96 57 42 40 66)(19 67 33 43 58 89)(20 90 59 44 34 68)(21 69 35 45 60 91)(22 92 61 46 36 70)(23 71 37 47 62 93)(24 94 63 48 38 72)
(1 70)(2 93)(3 72)(4 95)(5 66)(6 89)(7 68)(8 91)(9 21)(10 36)(11 23)(12 38)(13 17)(14 40)(15 19)(16 34)(18 81)(20 83)(22 85)(24 87)(25 45)(26 92)(27 47)(28 94)(29 41)(30 96)(31 43)(32 90)(33 50)(35 52)(37 54)(39 56)(42 74)(44 76)(46 78)(48 80)(49 57)(51 59)(53 61)(55 63)(58 82)(60 84)(62 86)(64 88)(65 73)(67 75)(69 77)(71 79)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 47)(18 42)(19 45)(20 48)(21 43)(22 46)(23 41)(24 44)(25 31)(27 29)(28 32)(33 91)(34 94)(35 89)(36 92)(37 95)(38 90)(39 93)(40 96)(50 52)(51 55)(54 56)(57 66)(58 69)(59 72)(60 67)(61 70)(62 65)(63 68)(64 71)(73 79)(75 77)(76 80)(82 84)(83 87)(86 88)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 1 1 6 6 8 2 2 2 6 6 8 24 24 2 2 2 8 8 2 2 2 2 6 6 6 6 4 4 8 8 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 C4○D4 SD16 C4○D8 C4○D12 S3×D4 S3×D4 D8⋊3S3 S3×SD16 kernel D6⋊SD16 C12.Q8 C2.Dic12 C3×D4⋊C4 C4.D12 S3×C2×C8 C2×D4.S3 D6⋊3D4 D4⋊C4 C3⋊C8 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C2×D4 C12 D6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 4 4 4 1 1 2 2

Matrix representation of D6⋊SD16 in GL4(𝔽73) generated by

 65 0 0 0 66 9 0 0 0 0 72 0 0 0 0 72
,
 57 18 0 0 71 16 0 0 0 0 46 34 0 0 43 27
,
 48 19 0 0 17 25 0 0 0 0 0 33 0 0 31 61
,
 1 0 0 0 18 72 0 0 0 0 1 0 0 0 66 72
G:=sub<GL(4,GF(73))| [65,66,0,0,0,9,0,0,0,0,72,0,0,0,0,72],[57,71,0,0,18,16,0,0,0,0,46,43,0,0,34,27],[48,17,0,0,19,25,0,0,0,0,0,31,0,0,33,61],[1,18,0,0,0,72,0,0,0,0,1,66,0,0,0,72] >;

D6⋊SD16 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm SD}_{16}
% in TeX

G:=Group("D6:SD16");
// GroupNames label

G:=SmallGroup(192,337);
// by ID

G=gap.SmallGroup(192,337);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,297,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^4*b,d*b*d=a^3*b,d*c*d=c^3>;
// generators/relations

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