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## G = Q16.Dic3order 192 = 26·3

### 2nd non-split extension by Q16 of Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — Q16.Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C24.C4 — Q16.Dic3
 Lower central C3 — C6 — C12 — C24 — Q16.Dic3
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for Q16.Dic3
G = < a,b,c,d | a8=1, b2=c6=a4, d2=a4c3, bab-1=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a3b, dcd-1=c5 >

Character table of Q16.Dic3

 class 1 2A 2B 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 8E 12A 12B 12C 12D 12E 12F 16A 16B 16C 16D 24A 24B 24C 24D size 1 1 2 2 2 2 8 8 2 2 2 2 2 4 24 24 4 4 8 8 8 8 12 12 12 12 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 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 -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 -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 1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -i i -1 1 -1 -1 1 1 -i i -i i 1 -1 1 -1 linear of order 4 ρ6 1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 -i i -1 1 1 1 -1 -1 i -i i -i 1 -1 1 -1 linear of order 4 ρ7 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 i -i -1 1 -1 -1 1 1 i -i i -i 1 -1 1 -1 linear of order 4 ρ8 1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 i -i -1 1 1 1 -1 -1 -i i -i i 1 -1 1 -1 linear of order 4 ρ9 2 2 2 -1 2 2 -2 -2 -1 -1 -1 2 2 2 0 0 -1 -1 1 1 1 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ10 2 2 -2 2 -2 2 0 0 -2 -2 2 2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 -2 2 -2 2 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 0 0 2 2 2 -2 -2 -2 0 0 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ12 2 2 2 -1 2 2 2 2 -1 -1 -1 2 2 2 0 0 -1 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 -2 2 2 -2 0 0 -2 -2 2 0 0 0 0 0 2 -2 0 0 0 0 -√2 -√2 √2 √2 0 0 0 0 orthogonal lifted from D8 ρ14 2 2 -2 2 2 -2 0 0 -2 -2 2 0 0 0 0 0 2 -2 0 0 0 0 √2 √2 -√2 -√2 0 0 0 0 orthogonal lifted from D8 ρ15 2 2 -2 -1 -2 2 -2 2 1 1 -1 -2 -2 2 0 0 1 -1 1 1 -1 -1 0 0 0 0 -1 1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ16 2 2 -2 -1 -2 2 2 -2 1 1 -1 -2 -2 2 0 0 1 -1 -1 -1 1 1 0 0 0 0 -1 1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ17 2 2 -2 -1 -2 2 0 0 1 1 -1 2 2 -2 0 0 1 -1 √-3 -√-3 √-3 -√-3 0 0 0 0 1 -1 1 -1 complex lifted from C3⋊D4 ρ18 2 2 2 -1 2 2 0 0 -1 -1 -1 -2 -2 -2 0 0 -1 -1 √-3 -√-3 -√-3 √-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ19 2 2 -2 -1 -2 2 0 0 1 1 -1 2 2 -2 0 0 1 -1 -√-3 √-3 -√-3 √-3 0 0 0 0 1 -1 1 -1 complex lifted from C3⋊D4 ρ20 2 2 2 -1 2 2 0 0 -1 -1 -1 -2 -2 -2 0 0 -1 -1 -√-3 √-3 √-3 -√-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ21 2 2 2 2 -2 -2 0 0 2 2 2 0 0 0 0 0 -2 -2 0 0 0 0 √-2 -√-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ22 2 2 2 2 -2 -2 0 0 2 2 2 0 0 0 0 0 -2 -2 0 0 0 0 -√-2 √-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ23 4 4 -4 -2 4 -4 0 0 2 2 -2 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 4 4 -2 -4 -4 0 0 -2 -2 -2 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ25 4 -4 0 4 0 0 0 0 0 0 -4 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 symplectic lifted from C8.17D4, Schur index 2 ρ26 4 -4 0 4 0 0 0 0 0 0 -4 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 symplectic lifted from C8.17D4, Schur index 2 ρ27 4 -4 0 -2 0 0 0 0 2√-3 -2√-3 2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 -√-6 √2 √-6 -√2 complex faithful ρ28 4 -4 0 -2 0 0 0 0 -2√-3 2√-3 2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 -√-6 -√2 √-6 √2 complex faithful ρ29 4 -4 0 -2 0 0 0 0 -2√-3 2√-3 2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 √-6 √2 -√-6 -√2 complex faithful ρ30 4 -4 0 -2 0 0 0 0 2√-3 -2√-3 2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 0 0 √-6 -√2 -√-6 √2 complex faithful

Smallest permutation representation of Q16.Dic3
On 96 points
Generators in S96
(1 15 4 18 7 21 10 24)(2 16 5 19 8 22 11 13)(3 17 6 20 9 23 12 14)(25 44 28 47 31 38 34 41)(26 45 29 48 32 39 35 42)(27 46 30 37 33 40 36 43)(49 61 58 70 55 67 52 64)(50 62 59 71 56 68 53 65)(51 63 60 72 57 69 54 66)(73 88 82 85 79 94 76 91)(74 89 83 86 80 95 77 92)(75 90 84 87 81 96 78 93)
(1 34 7 28)(2 29 8 35)(3 36 9 30)(4 31 10 25)(5 26 11 32)(6 33 12 27)(13 48 19 42)(14 43 20 37)(15 38 21 44)(16 45 22 39)(17 40 23 46)(18 47 24 41)(49 85 55 91)(50 92 56 86)(51 87 57 93)(52 94 58 88)(53 89 59 95)(54 96 60 90)(61 82 67 76)(62 77 68 83)(63 84 69 78)(64 79 70 73)(65 74 71 80)(66 81 72 75)
(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 58 10 55 7 52 4 49)(2 51 11 60 8 57 5 54)(3 56 12 53 9 50 6 59)(13 69 22 66 19 63 16 72)(14 62 23 71 20 68 17 65)(15 67 24 64 21 61 18 70)(25 73 34 82 31 79 28 76)(26 78 35 75 32 84 29 81)(27 83 36 80 33 77 30 74)(37 86 46 95 43 92 40 89)(38 91 47 88 44 85 41 94)(39 96 48 93 45 90 42 87)

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

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

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

Matrix representation of Q16.Dic3 in GL4(𝔽7) generated by

 2 6 0 6 5 2 2 6 0 4 0 2 1 5 6 3
,
 0 0 1 2 0 2 1 0 0 2 5 0 3 6 1 0
,
 0 3 1 0 0 3 0 4 2 6 0 6 6 0 3 4
,
 4 4 3 2 0 6 4 4 0 2 3 1 1 4 6 1
G:=sub<GL(4,GF(7))| [2,5,0,1,6,2,4,5,0,2,0,6,6,6,2,3],[0,0,0,3,0,2,2,6,1,1,5,1,2,0,0,0],[0,0,2,6,3,3,6,0,1,0,0,3,0,4,6,4],[4,0,0,1,4,6,2,4,3,4,3,6,2,4,1,1] >;

Q16.Dic3 in GAP, Magma, Sage, TeX

Q_{16}.{\rm Dic}_3
% in TeX

G:=Group("Q16.Dic3");
// GroupNames label

G:=SmallGroup(192,124);
// by ID

G=gap.SmallGroup(192,124);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,387,184,675,794,80,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=1,b^2=c^6=a^4,d^2=a^4*c^3,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^3*b,d*c*d^-1=c^5>;
// generators/relations

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