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## G = SD16⋊Dic3order 192 = 26·3

### 1st semidirect product of SD16 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for SD16⋊Dic3
G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=a3, ac=ca, dad-1=a5, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 296 in 120 conjugacy classes, 57 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×SD16, C22×Dic3, C6×D4, C6×Q8, SD16⋊C4, C24⋊C4, C241C4, D4⋊Dic3, Q82Dic3, D4×Dic3, Q8×Dic3, C6×SD16, SD16⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C8⋊C22, C8.C22, S3×D4, D42S3, C22×Dic3, SD16⋊C4, Q83D6, D4.D6, D4×Dic3, SD16⋊Dic3

Smallest permutation representation of SD16⋊Dic3
On 96 points
Generators in S96
(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)(10 12)(11 15)(14 16)(17 21)(18 24)(20 22)(25 27)(26 30)(29 31)(34 36)(35 39)(38 40)(42 44)(43 47)(46 48)(49 53)(50 56)(52 54)(57 63)(59 61)(60 64)(65 69)(66 72)(68 70)(74 76)(75 79)(78 80)(82 84)(83 87)(86 88)(89 93)(90 96)(92 94)
(1 62 51 77 19 41)(2 63 52 78 20 42)(3 64 53 79 21 43)(4 57 54 80 22 44)(5 58 55 73 23 45)(6 59 56 74 24 46)(7 60 49 75 17 47)(8 61 50 76 18 48)(9 37 71 81 91 32)(10 38 72 82 92 25)(11 39 65 83 93 26)(12 40 66 84 94 27)(13 33 67 85 95 28)(14 34 68 86 96 29)(15 35 69 87 89 30)(16 36 70 88 90 31)
(1 32 77 71)(2 29 78 68)(3 26 79 65)(4 31 80 70)(5 28 73 67)(6 25 74 72)(7 30 75 69)(8 27 76 66)(9 51 81 41)(10 56 82 46)(11 53 83 43)(12 50 84 48)(13 55 85 45)(14 52 86 42)(15 49 87 47)(16 54 88 44)(17 35 60 89)(18 40 61 94)(19 37 62 91)(20 34 63 96)(21 39 64 93)(22 36 57 90)(23 33 58 95)(24 38 59 92)

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

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

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

36 conjugacy classes

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

36 irreducible representations

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

Matrix representation of SD16⋊Dic3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 63 44 44 0 0 39 68 0 44 0 0 5 68 0 0 0 0 34 68 44 44
,
 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 25 25 72 0 0 0 25 25 0 72
,
 0 72 0 0 0 0 1 1 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0 0 0 72 1 1
,
 46 46 0 0 0 0 0 27 0 0 0 0 0 0 8 39 0 0 0 0 47 65 0 0 0 0 8 0 39 65 0 0 47 39 26 34

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,39,5,34,0,0,63,68,68,68,0,0,44,0,0,44,0,0,44,44,0,44],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,25,25,0,0,0,1,25,25,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,1,1,0,0,0,72,0,0,72,0,0,0,0,0,1,0,0,0,0,72,1],[46,0,0,0,0,0,46,27,0,0,0,0,0,0,8,47,8,47,0,0,39,65,0,39,0,0,0,0,39,26,0,0,0,0,65,34] >;

SD16⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(192,723);
// by ID

G=gap.SmallGroup(192,723);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,219,184,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=c^3,b*a*b=a^3,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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