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## G = Dic3⋊7SD16order 192 = 26·3

### 2nd semidirect product of Dic3 and SD16 acting through Inn(Dic3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Dic3⋊7SD16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — Dic3⋊5D4 — Dic3⋊7SD16
 Lower central C3 — C6 — C12 — Dic3⋊7SD16
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Dic37SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c3 >

Subgroups: 344 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, 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, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, Q82S3, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C6×Q8, C4×SD16, C12.Q8, C8×Dic3, C2.D24, C3×Q8⋊C4, Dic35D4, C2×Q82S3, Q8×Dic3, Dic37SD16
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, D42S3, C4×SD16, Dic34D4, S3×SD16, D24⋊C2, Dic37SD16

Smallest permutation representation of Dic37SD16
On 96 points
Generators in S96
(1 78 42 20 62 66)(2 67 63 21 43 79)(3 80 44 22 64 68)(4 69 57 23 45 73)(5 74 46 24 58 70)(6 71 59 17 47 75)(7 76 48 18 60 72)(8 65 61 19 41 77)(9 33 90 53 81 27)(10 28 82 54 91 34)(11 35 92 55 83 29)(12 30 84 56 93 36)(13 37 94 49 85 31)(14 32 86 50 95 38)(15 39 96 51 87 25)(16 26 88 52 89 40)
(1 10 20 54)(2 11 21 55)(3 12 22 56)(4 13 23 49)(5 14 24 50)(6 15 17 51)(7 16 18 52)(8 9 19 53)(25 47 96 71)(26 48 89 72)(27 41 90 65)(28 42 91 66)(29 43 92 67)(30 44 93 68)(31 45 94 69)(32 46 95 70)(33 61 81 77)(34 62 82 78)(35 63 83 79)(36 64 84 80)(37 57 85 73)(38 58 86 74)(39 59 87 75)(40 60 88 76)
(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 5)(2 8)(4 6)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(25 37)(26 40)(27 35)(28 38)(29 33)(30 36)(31 39)(32 34)(41 63)(42 58)(43 61)(44 64)(45 59)(46 62)(47 57)(48 60)(49 51)(50 54)(53 55)(65 79)(66 74)(67 77)(68 80)(69 75)(70 78)(71 73)(72 76)(81 92)(82 95)(83 90)(84 93)(85 96)(86 91)(87 94)(88 89)

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

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

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

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 2 4 4 4 4 type + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 SD16 C4○D4 C4×S3 C4○D8 D4⋊2S3 S3×D4 S3×SD16 D24⋊C2 kernel Dic3⋊7SD16 C12.Q8 C8×Dic3 C2.D24 C3×Q8⋊C4 Dic3⋊5D4 C2×Q8⋊2S3 Q8×Dic3 Q8⋊2S3 Q8⋊C4 C2×Dic3 C4⋊C4 C2×C8 C2×Q8 Dic3 C12 Q8 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 4 2 4 4 1 1 2 2

Matrix representation of Dic37SD16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 72 0 0 0 0 72 0
,
 67 6 0 0 0 0 67 67 0 0 0 0 0 0 0 2 0 0 0 0 36 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,0,36,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic37SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_7{\rm SD}_{16}
% in TeX

G:=Group("Dic3:7SD16");
// GroupNames label

G:=SmallGroup(192,347);
// by ID

G=gap.SmallGroup(192,347);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,120,135,184,570,297,136,6278]);
// Polycyclic

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

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