metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D12, C12⋊13SD16, C42.57D6, (C4×Q8)⋊7S3, (C3×Q8)⋊8D4, (Q8×C12)⋊3C2, C4⋊C4.253D6, C3⋊3(C4⋊SD16), C12⋊C8⋊26C2, C12.20(C2×D4), C4.16(C2×D12), (C2×C12).66D4, C4⋊3(Q8⋊2S3), C4⋊D12.6C2, C6.D8⋊32C2, (C2×Q8).184D6, C6.70(C2×SD16), C4.12(C4○D12), C12.60(C4○D4), C6.67(C4⋊D4), (C4×C12).98C22, C2.10(D4⋊D6), C2.15(C12⋊7D4), C6.112(C8⋊C22), (C2×C12).347C23, (C2×D12).95C22, (C6×Q8).195C22, (C2×Q8⋊2S3)⋊7C2, (C2×C6).478(C2×D4), C2.7(C2×Q8⋊2S3), (C2×C3⋊C8).101C22, (C2×C4).249(C3⋊D4), (C3×C4⋊C4).284C22, (C2×C4).447(C22×S3), C22.155(C2×C3⋊D4), SmallGroup(192,586)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Subgroups: 456 in 128 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C12, C12, D6, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C3⋊C8, D12, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C2×C3⋊C8, Q8⋊2S3, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×D12, C2×D12, C6×Q8, C4⋊SD16, C12⋊C8, C6.D8, C4⋊D12, C2×Q8⋊2S3, Q8×C12, Q8⋊2D12
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C8⋊C22, Q8⋊2S3, C2×D12, C4○D12, C2×C3⋊D4, C4⋊SD16, C12⋊7D4, C2×Q8⋊2S3, D4⋊D6, Q8⋊2D12
►On 96 points
(1 45 69 78)(2 46 70 79)(3 47 71 80)(4 48 72 81)(5 37 61 82)(6 38 62 83)(7 39 63 84)(8 40 64 73)(9 41 65 74)(10 42 66 75)(11 43 67 76)(12 44 68 77)(13 27 56 89)(14 28 57 90)(15 29 58 91)(16 30 59 92)(17 31 60 93)(18 32 49 94)(19 33 50 95)(20 34 51 96)(21 35 52 85)(22 36 53 86)(23 25 54 87)(24 26 55 88)
(1 93 69 31)(2 94 70 32)(3 95 71 33)(4 96 72 34)(5 85 61 35)(6 86 62 36)(7 87 63 25)(8 88 64 26)(9 89 65 27)(10 90 66 28)(11 91 67 29)(12 92 68 30)(13 41 56 74)(14 42 57 75)(15 43 58 76)(16 44 59 77)(17 45 60 78)(18 46 49 79)(19 47 50 80)(20 48 51 81)(21 37 52 82)(22 38 53 83)(23 39 54 84)(24 40 55 73)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 96)(14 95)(15 94)(16 93)(17 92)(18 91)(19 90)(20 89)(21 88)(22 87)(23 86)(24 85)(25 53)(26 52)(27 51)(28 50)(29 49)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 73)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)
G:=sub<Sym(96)| (1,45,69,78)(2,46,70,79)(3,47,71,80)(4,48,72,81)(5,37,61,82)(6,38,62,83)(7,39,63,84)(8,40,64,73)(9,41,65,74)(10,42,66,75)(11,43,67,76)(12,44,68,77)(13,27,56,89)(14,28,57,90)(15,29,58,91)(16,30,59,92)(17,31,60,93)(18,32,49,94)(19,33,50,95)(20,34,51,96)(21,35,52,85)(22,36,53,86)(23,25,54,87)(24,26,55,88), (1,93,69,31)(2,94,70,32)(3,95,71,33)(4,96,72,34)(5,85,61,35)(6,86,62,36)(7,87,63,25)(8,88,64,26)(9,89,65,27)(10,90,66,28)(11,91,67,29)(12,92,68,30)(13,41,56,74)(14,42,57,75)(15,43,58,76)(16,44,59,77)(17,45,60,78)(18,46,49,79)(19,47,50,80)(20,48,51,81)(21,37,52,82)(22,38,53,83)(23,39,54,84)(24,40,55,73), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,53)(26,52)(27,51)(28,50)(29,49)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,73)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)>;
G:=Group( (1,45,69,78)(2,46,70,79)(3,47,71,80)(4,48,72,81)(5,37,61,82)(6,38,62,83)(7,39,63,84)(8,40,64,73)(9,41,65,74)(10,42,66,75)(11,43,67,76)(12,44,68,77)(13,27,56,89)(14,28,57,90)(15,29,58,91)(16,30,59,92)(17,31,60,93)(18,32,49,94)(19,33,50,95)(20,34,51,96)(21,35,52,85)(22,36,53,86)(23,25,54,87)(24,26,55,88), (1,93,69,31)(2,94,70,32)(3,95,71,33)(4,96,72,34)(5,85,61,35)(6,86,62,36)(7,87,63,25)(8,88,64,26)(9,89,65,27)(10,90,66,28)(11,91,67,29)(12,92,68,30)(13,41,56,74)(14,42,57,75)(15,43,58,76)(16,44,59,77)(17,45,60,78)(18,46,49,79)(19,47,50,80)(20,48,51,81)(21,37,52,82)(22,38,53,83)(23,39,54,84)(24,40,55,73), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,53)(26,52)(27,51)(28,50)(29,49)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,73)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69) );
G=PermutationGroup([[(1,45,69,78),(2,46,70,79),(3,47,71,80),(4,48,72,81),(5,37,61,82),(6,38,62,83),(7,39,63,84),(8,40,64,73),(9,41,65,74),(10,42,66,75),(11,43,67,76),(12,44,68,77),(13,27,56,89),(14,28,57,90),(15,29,58,91),(16,30,59,92),(17,31,60,93),(18,32,49,94),(19,33,50,95),(20,34,51,96),(21,35,52,85),(22,36,53,86),(23,25,54,87),(24,26,55,88)], [(1,93,69,31),(2,94,70,32),(3,95,71,33),(4,96,72,34),(5,85,61,35),(6,86,62,36),(7,87,63,25),(8,88,64,26),(9,89,65,27),(10,90,66,28),(11,91,67,29),(12,92,68,30),(13,41,56,74),(14,42,57,75),(15,43,58,76),(16,44,59,77),(17,45,60,78),(18,46,49,79),(19,47,50,80),(20,48,51,81),(21,37,52,82),(22,38,53,83),(23,39,54,84),(24,40,55,73)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,96),(14,95),(15,94),(16,93),(17,92),(18,91),(19,90),(20,89),(21,88),(22,87),(23,86),(24,85),(25,53),(26,52),(27,51),(28,50),(29,49),(30,60),(31,59),(32,58),(33,57),(34,56),(35,55),(36,54),(37,73),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(61,64),(62,63),(65,72),(66,71),(67,70),(68,69)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 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 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | C4○D4 | C3⋊D4 | D12 | C4○D12 | C8⋊C22 | Q8⋊2S3 | D4⋊D6 |
| kernel | Q8⋊2D12 | C12⋊C8 | C6.D8 | C4⋊D12 | C2×Q8⋊2S3 | Q8×C12 | C4×Q8 | C2×C12 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C12 | C2×C4 | Q8 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of Q8⋊2D12 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 3 |
| 0 | 0 | 48 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 0 | 4 | 0 |
| 7 | 66 | 0 | 0 |
| 7 | 14 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 7 | 66 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 48 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,48,0,0,3,1],[72,0,0,0,0,72,0,0,0,0,0,4,0,0,18,0],[7,7,0,0,66,14,0,0,0,0,72,0,0,0,0,72],[7,59,0,0,66,66,0,0,0,0,72,48,0,0,0,1] >;
Q8⋊2D12 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_{12} % in TeX
G:=Group("Q8:2D12"); // GroupNames label
G:=SmallGroup(192,586);
// by ID
G=gap.SmallGroup(192,586);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations