metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.9D4, C6.5Q16, Q8⋊2Dic3, C6.8SD16, (C3×Q8)⋊1C4, C12.8(C2×C4), (C2×C4).40D6, (C2×C6).34D4, (C6×Q8).1C2, (C2×Q8).3S3, C3⋊3(Q8⋊C4), C4.2(C2×Dic3), C4.14(C3⋊D4), C4⋊Dic3.10C2, C2.3(C3⋊Q16), C6.16(C22⋊C4), (C2×C12).18C22, C2.3(Q8⋊2S3), C2.6(C6.D4), C22.18(C3⋊D4), (C2×C3⋊C8).5C2, SmallGroup(96,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2Dic3
G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >
Character table of Q8⋊2Dic3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 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 | i | -i | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | i | -1 | -1 | 1 | i | -i | i | -i | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | -1 | -1 | 1 | -i | i | -i | i | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | -1 | -1 | 1 | i | -i | i | -i | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2 | -2 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -√2 | √2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | √2 | -√2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | -√-3 | √-3 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | √-3 | -√-3 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -√-3 | √-3 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ22 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | √-3 | -√-3 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ23 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊2S3 |
| ρ24 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C3⋊Q16, Schur index 2 |
►Regular action on 96 points
Generators in S96
(1 28 23 10)(2 29 24 11)(3 30 19 12)(4 25 20 7)(5 26 21 8)(6 27 22 9)(13 94 85 80)(14 95 86 81)(15 96 87 82)(16 91 88 83)(17 92 89 84)(18 93 90 79)(31 41 44 54)(32 42 45 49)(33 37 46 50)(34 38 47 51)(35 39 48 52)(36 40 43 53)(55 66 69 73)(56 61 70 74)(57 62 71 75)(58 63 72 76)(59 64 67 77)(60 65 68 78)
(1 44 23 31)(2 45 24 32)(3 46 19 33)(4 47 20 34)(5 48 21 35)(6 43 22 36)(7 51 25 38)(8 52 26 39)(9 53 27 40)(10 54 28 41)(11 49 29 42)(12 50 30 37)(13 67 85 59)(14 68 86 60)(15 69 87 55)(16 70 88 56)(17 71 89 57)(18 72 90 58)(61 83 74 91)(62 84 75 92)(63 79 76 93)(64 80 77 94)(65 81 78 95)(66 82 73 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 56 4 59)(2 55 5 58)(3 60 6 57)(7 64 10 61)(8 63 11 66)(9 62 12 65)(13 54 16 51)(14 53 17 50)(15 52 18 49)(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 83 34 80)(32 82 35 79)(33 81 36 84)(37 86 40 89)(38 85 41 88)(39 90 42 87)(43 92 46 95)(44 91 47 94)(45 96 48 93)
G:=sub<Sym(96)| (1,28,23,10)(2,29,24,11)(3,30,19,12)(4,25,20,7)(5,26,21,8)(6,27,22,9)(13,94,85,80)(14,95,86,81)(15,96,87,82)(16,91,88,83)(17,92,89,84)(18,93,90,79)(31,41,44,54)(32,42,45,49)(33,37,46,50)(34,38,47,51)(35,39,48,52)(36,40,43,53)(55,66,69,73)(56,61,70,74)(57,62,71,75)(58,63,72,76)(59,64,67,77)(60,65,68,78), (1,44,23,31)(2,45,24,32)(3,46,19,33)(4,47,20,34)(5,48,21,35)(6,43,22,36)(7,51,25,38)(8,52,26,39)(9,53,27,40)(10,54,28,41)(11,49,29,42)(12,50,30,37)(13,67,85,59)(14,68,86,60)(15,69,87,55)(16,70,88,56)(17,71,89,57)(18,72,90,58)(61,83,74,91)(62,84,75,92)(63,79,76,93)(64,80,77,94)(65,81,78,95)(66,82,73,96), (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), (1,56,4,59)(2,55,5,58)(3,60,6,57)(7,64,10,61)(8,63,11,66)(9,62,12,65)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,83,34,80)(32,82,35,79)(33,81,36,84)(37,86,40,89)(38,85,41,88)(39,90,42,87)(43,92,46,95)(44,91,47,94)(45,96,48,93)>;
G:=Group( (1,28,23,10)(2,29,24,11)(3,30,19,12)(4,25,20,7)(5,26,21,8)(6,27,22,9)(13,94,85,80)(14,95,86,81)(15,96,87,82)(16,91,88,83)(17,92,89,84)(18,93,90,79)(31,41,44,54)(32,42,45,49)(33,37,46,50)(34,38,47,51)(35,39,48,52)(36,40,43,53)(55,66,69,73)(56,61,70,74)(57,62,71,75)(58,63,72,76)(59,64,67,77)(60,65,68,78), (1,44,23,31)(2,45,24,32)(3,46,19,33)(4,47,20,34)(5,48,21,35)(6,43,22,36)(7,51,25,38)(8,52,26,39)(9,53,27,40)(10,54,28,41)(11,49,29,42)(12,50,30,37)(13,67,85,59)(14,68,86,60)(15,69,87,55)(16,70,88,56)(17,71,89,57)(18,72,90,58)(61,83,74,91)(62,84,75,92)(63,79,76,93)(64,80,77,94)(65,81,78,95)(66,82,73,96), (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), (1,56,4,59)(2,55,5,58)(3,60,6,57)(7,64,10,61)(8,63,11,66)(9,62,12,65)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,83,34,80)(32,82,35,79)(33,81,36,84)(37,86,40,89)(38,85,41,88)(39,90,42,87)(43,92,46,95)(44,91,47,94)(45,96,48,93) );
G=PermutationGroup([[(1,28,23,10),(2,29,24,11),(3,30,19,12),(4,25,20,7),(5,26,21,8),(6,27,22,9),(13,94,85,80),(14,95,86,81),(15,96,87,82),(16,91,88,83),(17,92,89,84),(18,93,90,79),(31,41,44,54),(32,42,45,49),(33,37,46,50),(34,38,47,51),(35,39,48,52),(36,40,43,53),(55,66,69,73),(56,61,70,74),(57,62,71,75),(58,63,72,76),(59,64,67,77),(60,65,68,78)], [(1,44,23,31),(2,45,24,32),(3,46,19,33),(4,47,20,34),(5,48,21,35),(6,43,22,36),(7,51,25,38),(8,52,26,39),(9,53,27,40),(10,54,28,41),(11,49,29,42),(12,50,30,37),(13,67,85,59),(14,68,86,60),(15,69,87,55),(16,70,88,56),(17,71,89,57),(18,72,90,58),(61,83,74,91),(62,84,75,92),(63,79,76,93),(64,80,77,94),(65,81,78,95),(66,82,73,96)], [(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)], [(1,56,4,59),(2,55,5,58),(3,60,6,57),(7,64,10,61),(8,63,11,66),(9,62,12,65),(13,54,16,51),(14,53,17,50),(15,52,18,49),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,77,28,74),(26,76,29,73),(27,75,30,78),(31,83,34,80),(32,82,35,79),(33,81,36,84),(37,86,40,89),(38,85,41,88),(39,90,42,87),(43,92,46,95),(44,91,47,94),(45,96,48,93)]])
Q8⋊2Dic3 is a maximal subgroup of
Q8⋊2Dic6 Dic3.1Q16 Q8⋊3Dic6 (C2×C8).D6 Q8.3Dic6 (C2×Q8).36D6 Q8.4Dic6 Q8⋊C4⋊S3 S3×Q8⋊C4 (S3×Q8)⋊C4 Q8⋊7(C4×S3) C4⋊C4.150D6 D6.1SD16 D6.Q16 D6⋊C8.C2 C8⋊Dic3⋊C2 Q8⋊4Dic6 Q8⋊5Dic6 Q8.5Dic6 C4×Q8⋊2S3 C42.56D6 C4×C3⋊Q16 C42.59D6 (C2×Q8).49D6 (C2×C6).Q16 (C2×Q8).51D6 C3⋊C8⋊24D4 C3⋊C8⋊6D4 C3⋊C8.29D4 C3⋊C8.6D4 C42.61D6 C42.62D6 C42.213D6 D12.23D4 C12.9Q16 C42.77D6 C12⋊5SD16 C12⋊Q16 Dic3×SD16 Dic3⋊3SD16 SD16⋊Dic3 (C3×D4).D4 D6⋊8SD16 C24⋊14D4 D12⋊7D4 C24⋊8D4 Dic3×Q16 Dic3⋊3Q16 Q16⋊Dic3 (C2×Q16)⋊S3 D6⋊5Q16 D12.17D4 D6⋊3Q16 C24.36D4 (C6×Q8)⋊6C4 (C3×Q8)⋊13D4 (C2×C6)⋊8Q16 C4○D4⋊3Dic3 C4○D4⋊4Dic3 (C3×D4)⋊14D4 (C3×D4).32D4 Q8⋊2Dic9 Q8⋊Dic9 Dic6⋊Dic3 C6.Dic12 C62.117D4 C6.GL2(𝔽3) C30.Q16 C6.Dic20 Q8⋊2Dic15 Dic10⋊2Dic3
Q8⋊2Dic3 is a maximal quotient of
C12.C42 C12.26Q16 (C6×Q8)⋊C4 C12.5Q16 C12.10D8 Q8⋊2Dic9 Dic6⋊Dic3 C6.Dic12 C62.117D4 C30.Q16 C6.Dic20 Q8⋊2Dic15 Dic10⋊2Dic3
Matrix representation of Q8⋊2Dic3 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 65 | 64 | 0 | 0 |
| 0 | 64 | 8 | 0 | 0 |
| 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 60 | 43 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 46 | 0 | 0 | 0 | 0 |
| 0 | 29 | 6 | 0 | 0 |
| 0 | 6 | 44 | 0 | 0 |
| 0 | 0 | 0 | 5 | 18 |
| 0 | 0 | 0 | 23 | 68 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,65,64,0,0,0,64,8,0,0,0,0,0,30,60,0,0,0,13,43],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[46,0,0,0,0,0,29,6,0,0,0,6,44,0,0,0,0,0,5,23,0,0,0,18,68] >;
Q8⋊2Dic3 in GAP, Magma, Sage, TeX
Q_8\rtimes_2{\rm Dic}_3 % in TeX
G:=Group("Q8:2Dic3"); // GroupNames label
G:=SmallGroup(96,42);
// by ID
G=gap.SmallGroup(96,42);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,103,579,297,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^6=1,b^2=a^2,d^2=c^3,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8⋊2Dic3 in TeX
Character table of Q8⋊2Dic3 in TeX