metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊8Q8, Dic6⋊12D4, C42.173D6, C6.362- 1+4, C3⋊4(D4×Q8), C4⋊1(S3×Q8), C4⋊Q8⋊11S3, D6⋊7(C2×Q8), C12⋊3(C2×Q8), C12⋊Q8⋊44C2, C4.74(S3×D4), C4⋊C4.218D6, C12.72(C2×D4), D6⋊Q8⋊48C2, D6⋊3Q8⋊36C2, (C4×Dic6)⋊52C2, (C4×D12).26C2, (C2×Q8).170D6, C6.47(C22×Q8), (C2×C6).271C24, Dic3.30(C2×D4), C6.101(C22×D4), Dic3⋊5D4.13C2, (C2×C12).104C23, (C4×C12).212C22, D6⋊C4.152C22, (C6×Q8).138C22, (C2×D12).272C22, C4⋊Dic3.385C22, C22.292(S3×C23), Dic3⋊C4.166C22, (C22×S3).232C23, C2.37(Q8.15D6), (C2×Dic6).189C22, (C2×Dic3).142C23, (C4×Dic3).160C22, (C2×S3×Q8)⋊13C2, C2.74(C2×S3×D4), C2.30(C2×S3×Q8), (C3×C4⋊Q8)⋊13C2, (S3×C2×C4).145C22, (C3×C4⋊C4).214C22, (C2×C4).218(C22×S3), SmallGroup(192,1286)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊8Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c-1 >
Subgroups: 672 in 280 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, C2×D12, S3×Q8, C6×Q8, D4×Q8, C4×Dic6, C4×D12, C12⋊Q8, Dic3⋊5D4, D6⋊Q8, D6⋊3Q8, C3×C4⋊Q8, C2×S3×Q8, D12⋊8Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, C22×S3, C22×D4, C22×Q8, 2- 1+4, S3×D4, S3×Q8, S3×C23, D4×Q8, C2×S3×D4, C2×S3×Q8, Q8.15D6, D12⋊8Q8
►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 3)(4 12)(5 11)(6 10)(7 9)(13 17)(14 16)(18 24)(19 23)(20 22)(25 35)(26 34)(27 33)(28 32)(29 31)(37 47)(38 46)(39 45)(40 44)(41 43)(49 59)(50 58)(51 57)(52 56)(53 55)(61 67)(62 66)(63 65)(68 72)(69 71)(73 81)(74 80)(75 79)(76 78)(82 84)(85 87)(88 96)(89 95)(90 94)(91 93)
(1 82 59 14)(2 83 60 15)(3 84 49 16)(4 73 50 17)(5 74 51 18)(6 75 52 19)(7 76 53 20)(8 77 54 21)(9 78 55 22)(10 79 56 23)(11 80 57 24)(12 81 58 13)(25 93 71 43)(26 94 72 44)(27 95 61 45)(28 96 62 46)(29 85 63 47)(30 86 64 48)(31 87 65 37)(32 88 66 38)(33 89 67 39)(34 90 68 40)(35 91 69 41)(36 92 70 42)
(1 72 59 26)(2 67 60 33)(3 62 49 28)(4 69 50 35)(5 64 51 30)(6 71 52 25)(7 66 53 32)(8 61 54 27)(9 68 55 34)(10 63 56 29)(11 70 57 36)(12 65 58 31)(13 37 81 87)(14 44 82 94)(15 39 83 89)(16 46 84 96)(17 41 73 91)(18 48 74 86)(19 43 75 93)(20 38 76 88)(21 45 77 95)(22 40 78 90)(23 47 79 85)(24 42 80 92)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,35)(26,34)(27,33)(28,32)(29,31)(37,47)(38,46)(39,45)(40,44)(41,43)(49,59)(50,58)(51,57)(52,56)(53,55)(61,67)(62,66)(63,65)(68,72)(69,71)(73,81)(74,80)(75,79)(76,78)(82,84)(85,87)(88,96)(89,95)(90,94)(91,93), (1,82,59,14)(2,83,60,15)(3,84,49,16)(4,73,50,17)(5,74,51,18)(6,75,52,19)(7,76,53,20)(8,77,54,21)(9,78,55,22)(10,79,56,23)(11,80,57,24)(12,81,58,13)(25,93,71,43)(26,94,72,44)(27,95,61,45)(28,96,62,46)(29,85,63,47)(30,86,64,48)(31,87,65,37)(32,88,66,38)(33,89,67,39)(34,90,68,40)(35,91,69,41)(36,92,70,42), (1,72,59,26)(2,67,60,33)(3,62,49,28)(4,69,50,35)(5,64,51,30)(6,71,52,25)(7,66,53,32)(8,61,54,27)(9,68,55,34)(10,63,56,29)(11,70,57,36)(12,65,58,31)(13,37,81,87)(14,44,82,94)(15,39,83,89)(16,46,84,96)(17,41,73,91)(18,48,74,86)(19,43,75,93)(20,38,76,88)(21,45,77,95)(22,40,78,90)(23,47,79,85)(24,42,80,92)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,35)(26,34)(27,33)(28,32)(29,31)(37,47)(38,46)(39,45)(40,44)(41,43)(49,59)(50,58)(51,57)(52,56)(53,55)(61,67)(62,66)(63,65)(68,72)(69,71)(73,81)(74,80)(75,79)(76,78)(82,84)(85,87)(88,96)(89,95)(90,94)(91,93), (1,82,59,14)(2,83,60,15)(3,84,49,16)(4,73,50,17)(5,74,51,18)(6,75,52,19)(7,76,53,20)(8,77,54,21)(9,78,55,22)(10,79,56,23)(11,80,57,24)(12,81,58,13)(25,93,71,43)(26,94,72,44)(27,95,61,45)(28,96,62,46)(29,85,63,47)(30,86,64,48)(31,87,65,37)(32,88,66,38)(33,89,67,39)(34,90,68,40)(35,91,69,41)(36,92,70,42), (1,72,59,26)(2,67,60,33)(3,62,49,28)(4,69,50,35)(5,64,51,30)(6,71,52,25)(7,66,53,32)(8,61,54,27)(9,68,55,34)(10,63,56,29)(11,70,57,36)(12,65,58,31)(13,37,81,87)(14,44,82,94)(15,39,83,89)(16,46,84,96)(17,41,73,91)(18,48,74,86)(19,43,75,93)(20,38,76,88)(21,45,77,95)(22,40,78,90)(23,47,79,85)(24,42,80,92) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,17),(14,16),(18,24),(19,23),(20,22),(25,35),(26,34),(27,33),(28,32),(29,31),(37,47),(38,46),(39,45),(40,44),(41,43),(49,59),(50,58),(51,57),(52,56),(53,55),(61,67),(62,66),(63,65),(68,72),(69,71),(73,81),(74,80),(75,79),(76,78),(82,84),(85,87),(88,96),(89,95),(90,94),(91,93)], [(1,82,59,14),(2,83,60,15),(3,84,49,16),(4,73,50,17),(5,74,51,18),(6,75,52,19),(7,76,53,20),(8,77,54,21),(9,78,55,22),(10,79,56,23),(11,80,57,24),(12,81,58,13),(25,93,71,43),(26,94,72,44),(27,95,61,45),(28,96,62,46),(29,85,63,47),(30,86,64,48),(31,87,65,37),(32,88,66,38),(33,89,67,39),(34,90,68,40),(35,91,69,41),(36,92,70,42)], [(1,72,59,26),(2,67,60,33),(3,62,49,28),(4,69,50,35),(5,64,51,30),(6,71,52,25),(7,66,53,32),(8,61,54,27),(9,68,55,34),(10,63,56,29),(11,70,57,36),(12,65,58,31),(13,37,81,87),(14,44,82,94),(15,39,83,89),(16,46,84,96),(17,41,73,91),(18,48,74,86),(19,43,75,93),(20,38,76,88),(21,45,77,95),(22,40,78,90),(23,47,79,85),(24,42,80,92)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | 2- 1+4 | S3×D4 | S3×Q8 | Q8.15D6 |
| kernel | D12⋊8Q8 | C4×Dic6 | C4×D12 | C12⋊Q8 | Dic3⋊5D4 | D6⋊Q8 | D6⋊3Q8 | C3×C4⋊Q8 | C2×S3×Q8 | C4⋊Q8 | Dic6 | D12 | C42 | C4⋊C4 | C2×Q8 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of D12⋊8Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D12⋊8Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_8Q_8 % in TeX
G:=Group("D12:8Q8"); // GroupNames label
G:=SmallGroup(192,1286);
// by ID
G=gap.SmallGroup(192,1286);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,1571,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations