metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3.Q8, C4⋊C4.5S3, C2.5(S3×Q8), (C2×C4).10D6, C6.11(C2×Q8), C4⋊Dic3.6C2, C3⋊2(C42.C2), C6.11(C4○D4), Dic3⋊C4.5C2, (C2×C6).30C23, (C4×Dic3).9C2, C2.13(C4○D12), (C2×C12).55C22, C2.11(D4⋊2S3), C22.47(C22×S3), (C2×Dic3).9C22, (C3×C4⋊C4).6C2, SmallGroup(96,96)
Series: Derived ►Chief ►Lower central ►Upper central
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, D4⋊2S3, 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 |
►Regular action on 96 points
(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)]])
Dic3.Q8 is a maximal subgroup of
C6.102+ 1+4 C6.52- 1+4 C6.62- 1+4 C42.89D6 C42.93D6 C42.96D6 C42.102D6 C42.104D6 C42.105D6 C42.118D6 Dic6⋊10Q8 C42.232D6 C42.132D6 C42.134D6 C6.342+ 1+4 C6.702- 1+4 C6.442+ 1+4 C6.492+ 1+4 (Q8×Dic3)⋊C2 C6.752- 1+4 C6.152- 1+4 C6.1182+ 1+4 C6.522+ 1+4 C6.202- 1+4 C6.212- 1+4 C6.252- 1+4 C6.592+ 1+4 C4⋊C4.197D6 C6.802- 1+4 C6.812- 1+4 C6.632+ 1+4 C6.642+ 1+4 C6.662+ 1+4 C6.852- 1+4 Dic6⋊7Q8 C42.147D6 S3×C42.C2 C42.148D6 C42.150D6 C42.151D6 C42.154D6 C42.159D6 C42.189D6 C42.162D6 C42.163D6 C42.165D6 Dic6⋊8Q8 C42.174D6 C42.176D6 C42.180D6 Dic9.Q8 Dic3.Dic6 C62.16C23 C62.17C23 C62.37C23 C62.40C23 C62.233C23 Dic15.Q8 Dic15.2Q8 Dic3.Dic10 Dic3.2Dic10 Dic3.3Dic10 Dic15.4Q8 Dic15.3Q8
Dic3.Q8 is a maximal quotient of
C6.(C4×Q8) C3⋊(C42⋊8C4) C6.(C4×D4) C2.(C4×D12) C6.(C4⋊Q8) (C2×Dic3).9D4 (C2×C4).17D12 Dic3⋊(C4⋊C4) C6.67(C4×D4) C4⋊C4⋊5Dic3 (C2×C4).44D12 (C2×C12).54D4 (C2×Dic3).Q8 (C2×C12).288D4 Dic9.Q8 Dic3.Dic6 C62.16C23 C62.17C23 C62.37C23 C62.40C23 C62.233C23 Dic15.Q8 Dic15.2Q8 Dic3.Dic10 Dic3.2Dic10 Dic3.3Dic10 Dic15.4Q8 Dic15.3Q8
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
Export