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## G = Dic3×SD16order 192 = 26·3

### Direct product of Dic3 and SD16

Series: Derived Chief Lower central Upper central

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

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, bd=db, dcd=c3 >

Subgroups: 296 in 122 conjugacy classes, 59 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, 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, 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, C6.D4, C2×C24, C3×SD16, C22×Dic3, C6×D4, C6×Q8, C4×SD16, C8×Dic3, C8⋊Dic3, D4⋊Dic3, Q82Dic3, D4×Dic3, Q8×Dic3, C6×SD16, Dic3×SD16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, SD16, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C2×SD16, C4○D8, S3×D4, D42S3, C22×Dic3, C4×SD16, S3×SD16, Q8.7D6, D4×Dic3, Dic3×SD16

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

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

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

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

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 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 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 4 4 2 2 2 3 3 3 3 4 4 6 6 12 12 12 12 2 2 2 8 8 2 2 2 2 6 6 6 6 4 4 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 Dic3 D6 D6 SD16 C4○D4 C4○D8 D4⋊2S3 S3×D4 S3×SD16 Q8.7D6 kernel Dic3×SD16 C8×Dic3 C8⋊Dic3 D4⋊Dic3 Q8⋊2Dic3 D4×Dic3 Q8×Dic3 C6×SD16 C3×SD16 C2×SD16 C2×Dic3 C2×C8 SD16 C2×D4 C2×Q8 Dic3 C12 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 4 1 1 4 2 4 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 46 27 0 0 0 27
,
 0 6 0 0 61 61 0 0 0 0 1 0 0 0 0 1
,
 1 1 0 0 0 72 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,72,0],[1,0,0,0,0,1,0,0,0,0,46,0,0,0,27,27],[0,61,0,0,6,61,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,72,0,0,0,0,1,0,0,0,0,1] >;

Dic3×SD16 in GAP, Magma, Sage, TeX

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

G:=Group("Dic3xSD16");
// GroupNames label

G:=SmallGroup(192,720);
// by ID

G=gap.SmallGroup(192,720);
# by ID

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

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

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