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## G = Dic3.Q8order 96 = 25·3

### The non-split extension by Dic3 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic3.Q8
G = < a,b,c,d | a6=c4=1, b2=a3, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a3b, bd=db, dcd-1=a3c-1 >

Subgroups: 106 in 56 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, Dic3, Dic3, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Dic3, C2×C12, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, Dic3.Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C22×S3, C42.C2, C4○D12, D42S3, S3×Q8, Dic3.Q8

Character table of Dic3.Q8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 4 4 6 6 6 6 12 12 2 2 2 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 -1 1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 2 -1 -2 -2 -2 2 0 0 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -1 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 2 -1 2 2 -2 -2 0 0 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 2 2 -1 -2 -2 2 -2 0 0 0 0 0 0 -1 -1 -1 1 -1 1 1 1 -1 orthogonal lifted from D6 ρ13 2 -2 -2 2 2 0 0 0 0 -2 0 2 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -2 2 2 0 0 0 0 2 0 -2 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 2 -2 -2 2 -2i 2i 0 0 0 0 0 0 0 0 -2 2 -2 2i 0 0 -2i 0 0 complex lifted from C4○D4 ρ16 2 -2 2 -2 2 0 0 0 0 0 -2i 0 2i 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ17 2 -2 2 -2 2 0 0 0 0 0 2i 0 -2i 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 -2 2 2i -2i 0 0 0 0 0 0 0 0 -2 2 -2 -2i 0 0 2i 0 0 complex lifted from C4○D4 ρ19 2 2 -2 -2 -1 2i -2i 0 0 0 0 0 0 0 0 1 -1 1 i √-3 √3 -i -√3 -√-3 complex lifted from C4○D12 ρ20 2 2 -2 -2 -1 -2i 2i 0 0 0 0 0 0 0 0 1 -1 1 -i -√-3 √3 i -√3 √-3 complex lifted from C4○D12 ρ21 2 2 -2 -2 -1 2i -2i 0 0 0 0 0 0 0 0 1 -1 1 i -√-3 -√3 -i √3 √-3 complex lifted from C4○D12 ρ22 2 2 -2 -2 -1 -2i 2i 0 0 0 0 0 0 0 0 1 -1 1 -i √-3 -√3 i √3 -√-3 complex lifted from C4○D12 ρ23 4 -4 -4 4 -2 0 0 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2 ρ24 4 -4 4 -4 -2 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2

Smallest permutation representation of Dic3.Q8
Regular action on 96 points
Generators in S96
(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 79 4 82)(2 84 5 81)(3 83 6 80)(7 20 10 23)(8 19 11 22)(9 24 12 21)(13 87 16 90)(14 86 17 89)(15 85 18 88)(25 91 28 94)(26 96 29 93)(27 95 30 92)(31 59 34 56)(32 58 35 55)(33 57 36 60)(37 62 40 65)(38 61 41 64)(39 66 42 63)(43 71 46 68)(44 70 47 67)(45 69 48 72)(49 73 52 76)(50 78 53 75)(51 77 54 74)
(1 20 17 25)(2 21 18 26)(3 22 13 27)(4 23 14 28)(5 24 15 29)(6 19 16 30)(7 89 94 79)(8 90 95 80)(9 85 96 81)(10 86 91 82)(11 87 92 83)(12 88 93 84)(31 43 42 53)(32 44 37 54)(33 45 38 49)(34 46 39 50)(35 47 40 51)(36 48 41 52)(55 70 65 74)(56 71 66 75)(57 72 61 76)(58 67 62 77)(59 68 63 78)(60 69 64 73)
(1 58 17 62)(2 57 18 61)(3 56 13 66)(4 55 14 65)(5 60 15 64)(6 59 16 63)(7 54 94 44)(8 53 95 43)(9 52 96 48)(10 51 91 47)(11 50 92 46)(12 49 93 45)(19 75 30 71)(20 74 25 70)(21 73 26 69)(22 78 27 68)(23 77 28 67)(24 76 29 72)(31 87 42 83)(32 86 37 82)(33 85 38 81)(34 90 39 80)(35 89 40 79)(36 88 41 84)

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

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

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

Matrix representation of Dic3.Q8 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 9 0 0 0 0 0 2 3 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 5 1 0 0 0 0 0 8 0 0 0 0 0 0 11 6 0 0 0 0 6 2 0 0 0 0 0 0 5 11 0 0 0 0 0 8
,
 1 0 0 0 0 0 3 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 10 0 0 0 0 5 12
,
 1 8 0 0 0 0 0 12 0 0 0 0 0 0 11 6 0 0 0 0 6 2 0 0 0 0 0 0 5 0 0 0 0 0 0 5

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,2,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,1,8,0,0,0,0,0,0,11,6,0,0,0,0,6,2,0,0,0,0,0,0,5,0,0,0,0,0,11,8],[1,3,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,10,12],[1,0,0,0,0,0,8,12,0,0,0,0,0,0,11,6,0,0,0,0,6,2,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;

Dic3.Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_3.Q_8
% in TeX

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

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

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

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,218,188,86,2309]);
// Polycyclic

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

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