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## G = Dic3.SD16order 192 = 26·3

### 1st non-split extension by Dic3 of SD16 acting via SD16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic3.SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C12⋊3D4 — Dic3.SD16
 Lower central C3 — C6 — C2×C12 — Dic3.SD16
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic3.SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=a3c3 >

Subgroups: 424 in 118 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, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×C8, D4⋊C4, D4⋊C4, C41D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C2×D12, C2×C3⋊D4, C6×D4, C4.4D8, C6.D8, C8×Dic3, C2.D24, D4⋊Dic3, C3×D4⋊C4, C12⋊Q8, C123D4, Dic3.SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, SD16, C2×D4, C4○D4, C22×S3, C4.4D4, C2×D8, C2×SD16, C4○D12, S3×D4, D42S3, C4.4D8, C23.11D6, S3×D8, S3×SD16, Dic3.SD16

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 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 3 4 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 8 24 2 2 2 6 6 6 6 8 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 4 4 4 4 type + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 D8 SD16 C4○D4 C4○D12 D4⋊2S3 S3×D4 S3×D8 S3×SD16 kernel Dic3.SD16 C6.D8 C8×Dic3 C2.D24 D4⋊Dic3 C3×D4⋊C4 C12⋊Q8 C12⋊3D4 D4⋊C4 C2×Dic3 C4⋊C4 C2×C8 C2×D4 Dic3 Dic3 C12 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 4 4 4 4 1 1 2 2

Matrix representation of Dic3.SD16 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 16 65 0 0 0 0 23 57
,
 57 57 0 0 0 0 16 57 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 0 46
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 57 8 0 0 0 0 32 16

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,16,23,0,0,0,0,65,57],[57,16,0,0,0,0,57,57,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,57,32,0,0,0,0,8,16] >;

Dic3.SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(192,319);
// by ID

G=gap.SmallGroup(192,319);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,422,135,100,570,297,136,6278]);
// Polycyclic

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

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