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## G = Dic3⋊SD16order 192 = 26·3

### 1st semidirect product of Dic3 and SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic3⋊SD16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — Dic3⋊5D4 — Dic3⋊SD16
 Lower central C3 — C6 — C2×C12 — Dic3⋊SD16
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Dic3⋊SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=cac-1=dad=a-1, cbc-1=a3b, bd=db, dcd=c3 >

Subgroups: 376 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, 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, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, Q8⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C24⋊C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, Q82S3, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C6×Q8, D4.D4, C6.SD16, Dic3⋊C8, C3×Q8⋊C4, Dic35D4, C2×C24⋊C2, C2×Q82S3, Dic3⋊Q8, Dic3⋊SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C22×S3, C4⋊D4, C2×SD16, C8.C22, C4○D12, S3×D4, D4.D4, Dic3⋊D4, S3×SD16, Q16⋊S3, Dic3⋊SD16

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 SD16 C4○D4 C4○D12 C8.C22 S3×D4 S3×D4 S3×SD16 Q16⋊S3 kernel Dic3⋊SD16 C6.SD16 Dic3⋊C8 C3×Q8⋊C4 Dic3⋊5D4 C2×C24⋊C2 C2×Q8⋊2S3 Dic3⋊Q8 Q8⋊C4 D12 C2×Dic3 C4⋊C4 C2×C8 C2×Q8 Dic3 C12 C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 4 2 4 1 1 1 2 2

Matrix representation of Dic3⋊SD16 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 1 72 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 0 27 0 0 27 0
,
 0 55 0 0 4 61 0 0 0 0 13 30 0 0 43 60
,
 1 0 0 0 25 72 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,72,0],[1,0,0,0,0,1,0,0,0,0,0,27,0,0,27,0],[0,4,0,0,55,61,0,0,0,0,13,43,0,0,30,60],[1,25,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;

Dic3⋊SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(192,377);
// by ID

G=gap.SmallGroup(192,377);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,184,1684,851,438,102,6278]);
// Polycyclic

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

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