metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.6D12, C42.58D6, (C4×Q8)⋊8S3, (Q8×C12)⋊4C2, C4⋊C4.254D6, C12⋊C8⋊27C2, C12.21(C2×D4), C4.17(C2×D12), (C2×C12).67D4, (C3×Q8).18D4, C6.93(C4○D8), (C2×Q8).185D6, C3⋊4(Q8.D4), C12.61(C4○D4), C4.13(C4○D12), C6.SD16⋊32C2, C6.68(C4⋊D4), (C4×C12).99C22, C42⋊7S3.6C2, C6.D8.11C2, C2.16(C12⋊7D4), (C2×C12).348C23, (C2×D12).96C22, C6.88(C8.C22), (C6×Q8).196C22, C2.13(Q8.13D6), C2.9(Q8.11D6), (C2×Dic6).101C22, (C2×C3⋊Q16)⋊7C2, (C2×C6).479(C2×D4), (C2×C3⋊C8).102C22, (C2×Q8⋊2S3).5C2, (C2×C4).222(C3⋊D4), (C3×C4⋊C4).285C22, (C2×C4).448(C22×S3), C22.156(C2×C3⋊D4), SmallGroup(192,587)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.6D12
G = < a,b,c,d | a4=c12=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >
Subgroups: 328 in 112 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C3⋊C8, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C2×C3⋊C8, D6⋊C4, Q8⋊2S3, C3⋊Q16, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×D12, C6×Q8, Q8.D4, C12⋊C8, C6.D8, C6.SD16, C42⋊7S3, C2×Q8⋊2S3, C2×C3⋊Q16, Q8×C12, Q8.6D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C4○D8, C8.C22, C2×D12, C4○D12, C2×C3⋊D4, Q8.D4, C12⋊7D4, Q8.11D6, Q8.13D6, Q8.6D12
►On 96 points
(1 74 71 29)(2 75 72 30)(3 76 61 31)(4 77 62 32)(5 78 63 33)(6 79 64 34)(7 80 65 35)(8 81 66 36)(9 82 67 25)(10 83 68 26)(11 84 69 27)(12 73 70 28)(13 54 39 88)(14 55 40 89)(15 56 41 90)(16 57 42 91)(17 58 43 92)(18 59 44 93)(19 60 45 94)(20 49 46 95)(21 50 47 96)(22 51 48 85)(23 52 37 86)(24 53 38 87)
(1 93 71 59)(2 94 72 60)(3 95 61 49)(4 96 62 50)(5 85 63 51)(6 86 64 52)(7 87 65 53)(8 88 66 54)(9 89 67 55)(10 90 68 56)(11 91 69 57)(12 92 70 58)(13 81 39 36)(14 82 40 25)(15 83 41 26)(16 84 42 27)(17 73 43 28)(18 74 44 29)(19 75 45 30)(20 76 46 31)(21 77 47 32)(22 78 48 33)(23 79 37 34)(24 80 38 35)
(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 70 71 12)(2 11 72 69)(3 68 61 10)(4 9 62 67)(5 66 63 8)(6 7 64 65)(13 85 39 51)(14 50 40 96)(15 95 41 49)(16 60 42 94)(17 93 43 59)(18 58 44 92)(19 91 45 57)(20 56 46 90)(21 89 47 55)(22 54 48 88)(23 87 37 53)(24 52 38 86)(25 32 82 77)(26 76 83 31)(27 30 84 75)(28 74 73 29)(33 36 78 81)(34 80 79 35)
G:=sub<Sym(96)| (1,74,71,29)(2,75,72,30)(3,76,61,31)(4,77,62,32)(5,78,63,33)(6,79,64,34)(7,80,65,35)(8,81,66,36)(9,82,67,25)(10,83,68,26)(11,84,69,27)(12,73,70,28)(13,54,39,88)(14,55,40,89)(15,56,41,90)(16,57,42,91)(17,58,43,92)(18,59,44,93)(19,60,45,94)(20,49,46,95)(21,50,47,96)(22,51,48,85)(23,52,37,86)(24,53,38,87), (1,93,71,59)(2,94,72,60)(3,95,61,49)(4,96,62,50)(5,85,63,51)(6,86,64,52)(7,87,65,53)(8,88,66,54)(9,89,67,55)(10,90,68,56)(11,91,69,57)(12,92,70,58)(13,81,39,36)(14,82,40,25)(15,83,41,26)(16,84,42,27)(17,73,43,28)(18,74,44,29)(19,75,45,30)(20,76,46,31)(21,77,47,32)(22,78,48,33)(23,79,37,34)(24,80,38,35), (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), (1,70,71,12)(2,11,72,69)(3,68,61,10)(4,9,62,67)(5,66,63,8)(6,7,64,65)(13,85,39,51)(14,50,40,96)(15,95,41,49)(16,60,42,94)(17,93,43,59)(18,58,44,92)(19,91,45,57)(20,56,46,90)(21,89,47,55)(22,54,48,88)(23,87,37,53)(24,52,38,86)(25,32,82,77)(26,76,83,31)(27,30,84,75)(28,74,73,29)(33,36,78,81)(34,80,79,35)>;
G:=Group( (1,74,71,29)(2,75,72,30)(3,76,61,31)(4,77,62,32)(5,78,63,33)(6,79,64,34)(7,80,65,35)(8,81,66,36)(9,82,67,25)(10,83,68,26)(11,84,69,27)(12,73,70,28)(13,54,39,88)(14,55,40,89)(15,56,41,90)(16,57,42,91)(17,58,43,92)(18,59,44,93)(19,60,45,94)(20,49,46,95)(21,50,47,96)(22,51,48,85)(23,52,37,86)(24,53,38,87), (1,93,71,59)(2,94,72,60)(3,95,61,49)(4,96,62,50)(5,85,63,51)(6,86,64,52)(7,87,65,53)(8,88,66,54)(9,89,67,55)(10,90,68,56)(11,91,69,57)(12,92,70,58)(13,81,39,36)(14,82,40,25)(15,83,41,26)(16,84,42,27)(17,73,43,28)(18,74,44,29)(19,75,45,30)(20,76,46,31)(21,77,47,32)(22,78,48,33)(23,79,37,34)(24,80,38,35), (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), (1,70,71,12)(2,11,72,69)(3,68,61,10)(4,9,62,67)(5,66,63,8)(6,7,64,65)(13,85,39,51)(14,50,40,96)(15,95,41,49)(16,60,42,94)(17,93,43,59)(18,58,44,92)(19,91,45,57)(20,56,46,90)(21,89,47,55)(22,54,48,88)(23,87,37,53)(24,52,38,86)(25,32,82,77)(26,76,83,31)(27,30,84,75)(28,74,73,29)(33,36,78,81)(34,80,79,35) );
G=PermutationGroup([[(1,74,71,29),(2,75,72,30),(3,76,61,31),(4,77,62,32),(5,78,63,33),(6,79,64,34),(7,80,65,35),(8,81,66,36),(9,82,67,25),(10,83,68,26),(11,84,69,27),(12,73,70,28),(13,54,39,88),(14,55,40,89),(15,56,41,90),(16,57,42,91),(17,58,43,92),(18,59,44,93),(19,60,45,94),(20,49,46,95),(21,50,47,96),(22,51,48,85),(23,52,37,86),(24,53,38,87)], [(1,93,71,59),(2,94,72,60),(3,95,61,49),(4,96,62,50),(5,85,63,51),(6,86,64,52),(7,87,65,53),(8,88,66,54),(9,89,67,55),(10,90,68,56),(11,91,69,57),(12,92,70,58),(13,81,39,36),(14,82,40,25),(15,83,41,26),(16,84,42,27),(17,73,43,28),(18,74,44,29),(19,75,45,30),(20,76,46,31),(21,77,47,32),(22,78,48,33),(23,79,37,34),(24,80,38,35)], [(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)], [(1,70,71,12),(2,11,72,69),(3,68,61,10),(4,9,62,67),(5,66,63,8),(6,7,64,65),(13,85,39,51),(14,50,40,96),(15,95,41,49),(16,60,42,94),(17,93,43,59),(18,58,44,92),(19,91,45,57),(20,56,46,90),(21,89,47,55),(22,54,48,88),(23,87,37,53),(24,52,38,86),(25,32,82,77),(26,76,83,31),(27,30,84,75),(28,74,73,29),(33,36,78,81),(34,80,79,35)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C4○D8 | C4○D12 | C8.C22 | Q8.11D6 | Q8.13D6 |
| kernel | Q8.6D12 | C12⋊C8 | C6.D8 | C6.SD16 | C42⋊7S3 | C2×Q8⋊2S3 | C2×C3⋊Q16 | Q8×C12 | C4×Q8 | C2×C12 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C2×C4 | Q8 | C6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of Q8.6D12 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 48 | 72 |
| 43 | 13 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 0 | 0 | 12 | 18 |
| 0 | 0 | 69 | 61 |
| 7 | 66 | 0 | 0 |
| 7 | 14 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 7 | 66 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 18 | 27 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,48,0,0,3,72],[43,60,0,0,13,30,0,0,0,0,12,69,0,0,18,61],[7,7,0,0,66,14,0,0,0,0,46,0,0,0,0,46],[7,59,0,0,66,66,0,0,0,0,46,18,0,0,0,27] >;
Q8.6D12 in GAP, Magma, Sage, TeX
Q_8._6D_{12} % in TeX
G:=Group("Q8.6D12"); // GroupNames label
G:=SmallGroup(192,587);
// by ID
G=gap.SmallGroup(192,587);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=d^2=a^2,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=a^2*c^-1>;
// generators/relations