direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C12, C42.3C6, C4⋊C4.6C6, C12○3(C4⋊C4), C2.2(C6×Q8), C4.4(C2×C12), (C4×C12).9C2, (C2×Q8).7C6, C6.19(C2×Q8), C12.31(C2×C4), (C6×Q8).10C2, C6.40(C4○D4), C2.5(C22×C12), (C2×C6).74C23, C6.33(C22×C4), C22.8(C22×C6), (C2×C12).122C22, C4○3(C3×C4⋊C4), C12○3(C3×C4⋊C4), C2.3(C3×C4○D4), (C3×C4⋊C4).13C2, (C2×C4).16(C2×C6), SmallGroup(96,166)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C12
G = < a,b,c | a12=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 76 in 70 conjugacy classes, 64 normal (16 characteristic)
C1, C2 [×3], C3, C4 [×8], C4 [×3], C22, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C12 [×8], C12 [×3], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C2×C12, C2×C12 [×6], C3×Q8 [×4], C4×Q8, C4×C12 [×3], C3×C4⋊C4 [×3], C6×Q8, Q8×C12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], Q8 [×2], C23, C12 [×4], C2×C6 [×7], C22×C4, C2×Q8, C4○D4, C2×C12 [×6], C3×Q8 [×2], C22×C6, C4×Q8, C22×C12, C6×Q8, C3×C4○D4, Q8×C12
►Regular action 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 55 95 80)(2 56 96 81)(3 57 85 82)(4 58 86 83)(5 59 87 84)(6 60 88 73)(7 49 89 74)(8 50 90 75)(9 51 91 76)(10 52 92 77)(11 53 93 78)(12 54 94 79)(13 26 47 72)(14 27 48 61)(15 28 37 62)(16 29 38 63)(17 30 39 64)(18 31 40 65)(19 32 41 66)(20 33 42 67)(21 34 43 68)(22 35 44 69)(23 36 45 70)(24 25 46 71)
(1 48 95 14)(2 37 96 15)(3 38 85 16)(4 39 86 17)(5 40 87 18)(6 41 88 19)(7 42 89 20)(8 43 90 21)(9 44 91 22)(10 45 92 23)(11 46 93 24)(12 47 94 13)(25 78 71 53)(26 79 72 54)(27 80 61 55)(28 81 62 56)(29 82 63 57)(30 83 64 58)(31 84 65 59)(32 73 66 60)(33 74 67 49)(34 75 68 50)(35 76 69 51)(36 77 70 52)
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), (1,55,95,80)(2,56,96,81)(3,57,85,82)(4,58,86,83)(5,59,87,84)(6,60,88,73)(7,49,89,74)(8,50,90,75)(9,51,91,76)(10,52,92,77)(11,53,93,78)(12,54,94,79)(13,26,47,72)(14,27,48,61)(15,28,37,62)(16,29,38,63)(17,30,39,64)(18,31,40,65)(19,32,41,66)(20,33,42,67)(21,34,43,68)(22,35,44,69)(23,36,45,70)(24,25,46,71), (1,48,95,14)(2,37,96,15)(3,38,85,16)(4,39,86,17)(5,40,87,18)(6,41,88,19)(7,42,89,20)(8,43,90,21)(9,44,91,22)(10,45,92,23)(11,46,93,24)(12,47,94,13)(25,78,71,53)(26,79,72,54)(27,80,61,55)(28,81,62,56)(29,82,63,57)(30,83,64,58)(31,84,65,59)(32,73,66,60)(33,74,67,49)(34,75,68,50)(35,76,69,51)(36,77,70,52)>;
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), (1,55,95,80)(2,56,96,81)(3,57,85,82)(4,58,86,83)(5,59,87,84)(6,60,88,73)(7,49,89,74)(8,50,90,75)(9,51,91,76)(10,52,92,77)(11,53,93,78)(12,54,94,79)(13,26,47,72)(14,27,48,61)(15,28,37,62)(16,29,38,63)(17,30,39,64)(18,31,40,65)(19,32,41,66)(20,33,42,67)(21,34,43,68)(22,35,44,69)(23,36,45,70)(24,25,46,71), (1,48,95,14)(2,37,96,15)(3,38,85,16)(4,39,86,17)(5,40,87,18)(6,41,88,19)(7,42,89,20)(8,43,90,21)(9,44,91,22)(10,45,92,23)(11,46,93,24)(12,47,94,13)(25,78,71,53)(26,79,72,54)(27,80,61,55)(28,81,62,56)(29,82,63,57)(30,83,64,58)(31,84,65,59)(32,73,66,60)(33,74,67,49)(34,75,68,50)(35,76,69,51)(36,77,70,52) );
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)], [(1,55,95,80),(2,56,96,81),(3,57,85,82),(4,58,86,83),(5,59,87,84),(6,60,88,73),(7,49,89,74),(8,50,90,75),(9,51,91,76),(10,52,92,77),(11,53,93,78),(12,54,94,79),(13,26,47,72),(14,27,48,61),(15,28,37,62),(16,29,38,63),(17,30,39,64),(18,31,40,65),(19,32,41,66),(20,33,42,67),(21,34,43,68),(22,35,44,69),(23,36,45,70),(24,25,46,71)], [(1,48,95,14),(2,37,96,15),(3,38,85,16),(4,39,86,17),(5,40,87,18),(6,41,88,19),(7,42,89,20),(8,43,90,21),(9,44,91,22),(10,45,92,23),(11,46,93,24),(12,47,94,13),(25,78,71,53),(26,79,72,54),(27,80,61,55),(28,81,62,56),(29,82,63,57),(30,83,64,58),(31,84,65,59),(32,73,66,60),(33,74,67,49),(34,75,68,50),(35,76,69,51),(36,77,70,52)])
Q8×C12 is a maximal subgroup of
C12.26Q16 Q8⋊4Dic6 Q8⋊5Dic6 Q8.5Dic6 C42.210D6 C42.56D6 Q8⋊2D12 Q8.6D12 C42.59D6 C12⋊7Q16 Dic6⋊10Q8 C42.122D6 Q8⋊6Dic6 Q8⋊7Dic6 C42.125D6 C42.126D6 Q8⋊6D12 Q8⋊7D12 C42.232D6 D12⋊10Q8 C42.131D6 C42.132D6 C42.133D6 C42.134D6 C42.135D6 C42.136D6
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 6A | ··· | 6F | 12A | ··· | 12H | 12I | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | Q8 | C4○D4 | C3×Q8 | C3×C4○D4 |
| kernel | Q8×C12 | C4×C12 | C3×C4⋊C4 | C6×Q8 | C4×Q8 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C12 | C6 | C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 2 | 8 | 6 | 6 | 2 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of Q8×C12 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,10,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,0,8,0,0,8,0] >;
Q8×C12 in GAP, Magma, Sage, TeX
Q_8\times C_{12} % in TeX
G:=Group("Q8xC12"); // GroupNames label
G:=SmallGroup(96,166);
// by ID
G=gap.SmallGroup(96,166);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,151,338]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations