direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×Dic3, C3⋊3(C4×Q8), (C3×Q8)⋊3C4, C2.3(S3×Q8), (C2×C4).56D6, (C2×Q8).7S3, (C6×Q8).5C2, C6.16(C2×Q8), C12.14(C2×C4), C4.4(C2×Dic3), C6.35(C4○D4), C4⋊Dic3.12C2, (C2×C6).57C23, C6.26(C22×C4), (C4×Dic3).4C2, (C2×C12).39C22, C2.3(Q8⋊3S3), C2.7(C22×Dic3), C22.26(C22×S3), (C2×Dic3).40C22, SmallGroup(96,152)
Series: Derived ►Chief ►Lower central ►Upper central
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, Q8⋊3S3, 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 |
►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)]])
Q8×Dic3 is a maximal subgroup of
Dic3⋊7SD16 C3⋊Q16⋊C4 Dic3⋊4Q16 Q8⋊2Dic6 Q8⋊3Dic6 Q8.3Dic6 Q8.4Dic6 Q8⋊3(C4×S3) Dic3⋊5SD16 SD16⋊Dic3 (C3×Q8).D4 Dic3⋊3Q16 Q16⋊Dic3 (C2×Q16)⋊S3 Q8⋊6Dic6 Q8⋊7Dic6 C4×S3×Q8 C42.125D6 C4×Q8⋊3S3 C42.126D6 (Q8×Dic3)⋊C2 C4⋊C4.187D6 C6.152- 1+4 C6.1182+ 1+4 C6.212- 1+4 C6.232- 1+4 C6.772- 1+4 C6.242- 1+4 C42.139D6 C42.234D6 C42.143D6 C42.144D6 Dic6⋊8Q8 C42.241D6 C42.176D6 C42.177D6 C6.422- 1+4 C6.452- 1+4 C6.1442+ 1+4 C6.1072- 1+4 C6.1482+ 1+4 C62.13C23 Dic15⋊6Q8
Q8×Dic3 is a maximal quotient of
C4⋊C4⋊5Dic3 C4⋊C4⋊6Dic3 C42.210D6 (C6×Q8)⋊7C4 C62.13C23 Dic15⋊6Q8
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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