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## G = Q8×Dic3order 96 = 25·3

### Direct product of Q8 and Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×Dic3
G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 114 in 70 conjugacy classes, 51 normal (14 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C2×C6, C42, C4⋊C4, C2×Q8, C2×Dic3, C2×Dic3, C2×C12, C3×Q8, C4×Q8, C4×Dic3, C4⋊Dic3, C6×Q8, Q8×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, Dic3, D6, C22×C4, C2×Q8, C4○D4, C2×Dic3, C22×S3, C4×Q8, S3×Q8, Q83S3, C22×Dic3, Q8×Dic3

Character table of Q8×Dic3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 6 6 6 6 6 6 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 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 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 -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 -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 1 1 -1 -1 -1 -1 linear of order 2 ρ9 1 1 -1 -1 1 1 -1 1 -1 1 -1 -i -i i i i i -i i -i -i -1 -1 1 -1 1 1 1 -1 -1 linear of order 4 ρ10 1 1 -1 -1 1 1 1 -1 -1 -1 1 i i -i -i i i i -i -i -i -1 -1 1 1 -1 -1 1 1 -1 linear of order 4 ρ11 1 1 -1 -1 1 1 -1 1 -1 1 -1 i i -i -i -i -i i -i i i -1 -1 1 -1 1 1 1 -1 -1 linear of order 4 ρ12 1 1 -1 -1 1 1 1 -1 -1 -1 1 -i -i i i -i -i -i i i i -1 -1 1 1 -1 -1 1 1 -1 linear of order 4 ρ13 1 1 -1 -1 1 -1 -1 1 1 -1 1 -i -i i i -i i i -i i -i -1 -1 1 -1 1 -1 -1 1 1 linear of order 4 ρ14 1 1 -1 -1 1 -1 1 -1 1 1 -1 i i -i -i -i i -i i i -i -1 -1 1 1 -1 1 -1 -1 1 linear of order 4 ρ15 1 1 -1 -1 1 -1 -1 1 1 -1 1 i i -i -i i -i -i i -i i -1 -1 1 -1 1 -1 -1 1 1 linear of order 4 ρ16 1 1 -1 -1 1 -1 1 -1 1 1 -1 -i -i i i i -i i -i -i i -1 -1 1 1 -1 1 -1 -1 1 linear of order 4 ρ17 2 2 2 2 -1 -2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ18 2 2 2 2 -1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 2 2 -1 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ20 2 2 2 2 -1 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 -1 1 -1 orthogonal lifted from D6 ρ21 2 2 -2 -2 -1 -2 -2 2 2 -2 2 0 0 0 0 0 0 0 0 0 0 1 1 -1 1 -1 1 1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ22 2 2 -2 -2 -1 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 1 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ23 2 -2 2 -2 2 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ24 2 -2 2 -2 2 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ25 2 2 -2 -2 -1 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 1 -1 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ26 2 2 -2 -2 -1 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 1 1 -1 1 -1 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ27 2 -2 -2 2 2 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 -2 -2 2 2 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 4 -4 -4 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ30 4 -4 4 -4 -2 0 0 0 0 0 0 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

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

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

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

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

Matrix representation of Q8×Dic3 in GL4(𝔽13) generated by

 12 11 0 0 1 1 0 0 0 0 12 0 0 0 0 12
,
 5 0 0 0 8 8 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 1 12 0 0 1 0
,
 8 0 0 0 0 8 0 0 0 0 9 2 0 0 11 4
G:=sub<GL(4,GF(13))| [12,1,0,0,11,1,0,0,0,0,12,0,0,0,0,12],[5,8,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,1,0,0,12,0],[8,0,0,0,0,8,0,0,0,0,9,11,0,0,2,4] >;

Q8×Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xDic3");
// GroupNames label

G:=SmallGroup(96,152);
// by ID

G=gap.SmallGroup(96,152);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,188,86,2309]);
// Polycyclic

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

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