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## G = Dic3⋊Q8order 96 = 25·3

### 2nd semidirect product of Dic3 and 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 — C2×Q8

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

Subgroups: 130 in 68 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×Q8, C2×Q8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4⋊Q8, C4×Dic3, Dic3⋊C4, C2×Dic6, C6×Q8, Dic3⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C3⋊D4, C22×S3, C4⋊Q8, S3×Q8, C2×C3⋊D4, 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 2 -2 2 0 0 0 0 0 0 0 0 2 -2 -2 2 0 0 -2 0 0 orthogonal lifted from D4 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 D6 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 S3 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 2 -2 0 0 0 0 0 0 0 0 2 -2 -2 -2 0 0 2 0 0 orthogonal lifted from D4 ρ14 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 ρ15 2 2 -2 -2 2 0 0 0 0 0 -2 0 2 0 0 -2 2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 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 ρ17 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 ρ18 2 2 -2 -2 2 0 0 0 0 0 2 0 -2 0 0 -2 2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 2 -2 -1 -2 2 0 0 0 0 0 0 0 0 -1 1 1 -1 -√-3 -√-3 1 √-3 √-3 complex lifted from C3⋊D4 ρ20 2 -2 2 -2 -1 2 -2 0 0 0 0 0 0 0 0 -1 1 1 1 √-3 -√-3 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ21 2 -2 2 -2 -1 2 -2 0 0 0 0 0 0 0 0 -1 1 1 1 -√-3 √-3 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 -2 2 -2 -1 -2 2 0 0 0 0 0 0 0 0 -1 1 1 -1 √-3 √-3 1 -√-3 -√-3 complex lifted from C3⋊D4 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 S3×Q8, 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 85 16 88)(14 90 17 87)(15 89 18 86)(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 23 18 25)(2 24 13 26)(3 19 14 27)(4 20 15 28)(5 21 16 29)(6 22 17 30)(7 89 91 82)(8 90 92 83)(9 85 93 84)(10 86 94 79)(11 87 95 80)(12 88 96 81)(31 43 39 50)(32 44 40 51)(33 45 41 52)(34 46 42 53)(35 47 37 54)(36 48 38 49)(55 70 62 77)(56 71 63 78)(57 72 64 73)(58 67 65 74)(59 68 66 75)(60 69 61 76)
(1 35 18 37)(2 36 13 38)(3 31 14 39)(4 32 15 40)(5 33 16 41)(6 34 17 42)(7 74 91 67)(8 75 92 68)(9 76 93 69)(10 77 94 70)(11 78 95 71)(12 73 96 72)(19 50 27 43)(20 51 28 44)(21 52 29 45)(22 53 30 46)(23 54 25 47)(24 49 26 48)(55 86 62 79)(56 87 63 80)(57 88 64 81)(58 89 65 82)(59 90 66 83)(60 85 61 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)(13,85,16,88)(14,90,17,87)(15,89,18,86)(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), (1,23,18,25)(2,24,13,26)(3,19,14,27)(4,20,15,28)(5,21,16,29)(6,22,17,30)(7,89,91,82)(8,90,92,83)(9,85,93,84)(10,86,94,79)(11,87,95,80)(12,88,96,81)(31,43,39,50)(32,44,40,51)(33,45,41,52)(34,46,42,53)(35,47,37,54)(36,48,38,49)(55,70,62,77)(56,71,63,78)(57,72,64,73)(58,67,65,74)(59,68,66,75)(60,69,61,76), (1,35,18,37)(2,36,13,38)(3,31,14,39)(4,32,15,40)(5,33,16,41)(6,34,17,42)(7,74,91,67)(8,75,92,68)(9,76,93,69)(10,77,94,70)(11,78,95,71)(12,73,96,72)(19,50,27,43)(20,51,28,44)(21,52,29,45)(22,53,30,46)(23,54,25,47)(24,49,26,48)(55,86,62,79)(56,87,63,80)(57,88,64,81)(58,89,65,82)(59,90,66,83)(60,85,61,84)>;

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

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

Matrix representation of Dic3⋊Q8 in GL6(𝔽13)

 9 0 0 0 0 0 0 3 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 9 10 0 0 0 0 10 4 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 3 0 0 0 0 3 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,10,0,0,0,0,10,4,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3⋊Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(96,151);
// by ID

G=gap.SmallGroup(96,151);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,362,116,50,2309]);
// Polycyclic

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

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