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## G = Q8⋊2Dic3order 96 = 25·3

### 1st semidirect product of Q8 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8⋊2Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — Q8⋊2Dic3
 Lower central C3 — C6 — C12 — Q8⋊2Dic3
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for Q82Dic3
G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

Character table of Q82Dic3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 4 4 12 12 2 2 2 6 6 6 6 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 1 -1 1 1 -1 i -i -1 -1 1 -i i -i i -1 1 1 1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 1 -1 1 1 -1 -i i -1 -1 1 i -i i -i -1 1 1 1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 1 -1 1 -1 1 -i i -1 -1 1 -i i -i i -1 -1 -1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 -1 1 -1 1 i -i -1 -1 1 i -i i -i -1 -1 -1 1 1 1 linear of order 4 ρ9 2 2 2 2 -1 2 2 2 2 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 2 2 -2 0 0 0 0 -2 -2 2 0 0 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 -2 -2 0 0 0 0 2 2 2 0 0 0 0 -2 0 0 -2 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -1 2 2 -2 -2 0 0 -1 -1 -1 0 0 0 0 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 -2 -1 -2 2 2 -2 0 0 1 1 -1 0 0 0 0 1 -1 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 -2 -2 -1 -2 2 -2 2 0 0 1 1 -1 0 0 0 0 1 1 1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -2 2 2 0 0 0 0 0 0 -2 2 -2 -√2 √2 √2 -√2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ16 2 -2 -2 2 2 0 0 0 0 0 0 -2 2 -2 √2 -√2 -√2 √2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ17 2 2 -2 -2 -1 2 -2 0 0 0 0 1 1 -1 0 0 0 0 -1 -√-3 √-3 1 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 -1 2 -2 0 0 0 0 1 1 -1 0 0 0 0 -1 √-3 -√-3 1 -√-3 √-3 complex lifted from C3⋊D4 ρ19 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1 -√-3 √-3 1 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 -2 2 -2 2 0 0 0 0 0 0 2 -2 -2 √-2 √-2 -√-2 -√-2 0 0 0 0 0 0 complex lifted from SD16 ρ21 2 -2 2 -2 2 0 0 0 0 0 0 2 -2 -2 -√-2 -√-2 √-2 √-2 0 0 0 0 0 0 complex lifted from SD16 ρ22 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 1 √-3 -√-3 1 √-3 -√-3 complex lifted from C3⋊D4 ρ23 4 -4 4 -4 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ24 4 -4 -4 4 -2 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C3⋊Q16, Schur index 2

Smallest permutation representation of Q82Dic3
Regular action on 96 points
Generators in S96
(1 28 23 7)(2 29 24 8)(3 30 19 9)(4 25 20 10)(5 26 21 11)(6 27 22 12)(13 94 81 89)(14 95 82 90)(15 96 83 85)(16 91 84 86)(17 92 79 87)(18 93 80 88)(31 47 51 39)(32 48 52 40)(33 43 53 41)(34 44 54 42)(35 45 49 37)(36 46 50 38)(55 68 78 63)(56 69 73 64)(57 70 74 65)(58 71 75 66)(59 72 76 61)(60 67 77 62)
(1 44 23 42)(2 45 24 37)(3 46 19 38)(4 47 20 39)(5 48 21 40)(6 43 22 41)(7 54 28 34)(8 49 29 35)(9 50 30 36)(10 51 25 31)(11 52 26 32)(12 53 27 33)(13 67 81 62)(14 68 82 63)(15 69 83 64)(16 70 84 65)(17 71 79 66)(18 72 80 61)(55 90 78 95)(56 85 73 96)(57 86 74 91)(58 87 75 92)(59 88 76 93)(60 89 77 94)
(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 65 4 62)(2 64 5 61)(3 63 6 66)(7 57 10 60)(8 56 11 59)(9 55 12 58)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 81 34 84)(32 80 35 83)(33 79 36 82)(37 85 40 88)(38 90 41 87)(39 89 42 86)(43 92 46 95)(44 91 47 94)(45 96 48 93)

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

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

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

Matrix representation of Q82Dic3 in GL5(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 65 64 0 0 0 64 8 0 0 0 0 0 30 13 0 0 0 60 43
,
 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 1 0 0 0 72 0
,
 46 0 0 0 0 0 29 6 0 0 0 6 44 0 0 0 0 0 5 18 0 0 0 23 68

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,65,64,0,0,0,64,8,0,0,0,0,0,30,60,0,0,0,13,43],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[46,0,0,0,0,0,29,6,0,0,0,6,44,0,0,0,0,0,5,23,0,0,0,18,68] >;

Q82Dic3 in GAP, Magma, Sage, TeX

Q_8\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("Q8:2Dic3");
// GroupNames label

G:=SmallGroup(96,42);
// by ID

G=gap.SmallGroup(96,42);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,103,579,297,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^6=1,b^2=a^2,d^2=c^3,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

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