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, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C2×Q8, C2×C12, C2×C12, C3×Q8, C4×Q8, C4×C12, C3×C4⋊C4, C6×Q8, Q8×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, Q8, C23, C12, C2×C6, C22×C4, C2×Q8, C4○D4, C2×C12, C3×Q8, 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 48 83 21)(2 37 84 22)(3 38 73 23)(4 39 74 24)(5 40 75 13)(6 41 76 14)(7 42 77 15)(8 43 78 16)(9 44 79 17)(10 45 80 18)(11 46 81 19)(12 47 82 20)(25 94 72 57)(26 95 61 58)(27 96 62 59)(28 85 63 60)(29 86 64 49)(30 87 65 50)(31 88 66 51)(32 89 67 52)(33 90 68 53)(34 91 69 54)(35 92 70 55)(36 93 71 56)
(1 95 83 58)(2 96 84 59)(3 85 73 60)(4 86 74 49)(5 87 75 50)(6 88 76 51)(7 89 77 52)(8 90 78 53)(9 91 79 54)(10 92 80 55)(11 93 81 56)(12 94 82 57)(13 65 40 30)(14 66 41 31)(15 67 42 32)(16 68 43 33)(17 69 44 34)(18 70 45 35)(19 71 46 36)(20 72 47 25)(21 61 48 26)(22 62 37 27)(23 63 38 28)(24 64 39 29)
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), (1,48,83,21)(2,37,84,22)(3,38,73,23)(4,39,74,24)(5,40,75,13)(6,41,76,14)(7,42,77,15)(8,43,78,16)(9,44,79,17)(10,45,80,18)(11,46,81,19)(12,47,82,20)(25,94,72,57)(26,95,61,58)(27,96,62,59)(28,85,63,60)(29,86,64,49)(30,87,65,50)(31,88,66,51)(32,89,67,52)(33,90,68,53)(34,91,69,54)(35,92,70,55)(36,93,71,56), (1,95,83,58)(2,96,84,59)(3,85,73,60)(4,86,74,49)(5,87,75,50)(6,88,76,51)(7,89,77,52)(8,90,78,53)(9,91,79,54)(10,92,80,55)(11,93,81,56)(12,94,82,57)(13,65,40,30)(14,66,41,31)(15,67,42,32)(16,68,43,33)(17,69,44,34)(18,70,45,35)(19,71,46,36)(20,72,47,25)(21,61,48,26)(22,62,37,27)(23,63,38,28)(24,64,39,29)>;
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), (1,48,83,21)(2,37,84,22)(3,38,73,23)(4,39,74,24)(5,40,75,13)(6,41,76,14)(7,42,77,15)(8,43,78,16)(9,44,79,17)(10,45,80,18)(11,46,81,19)(12,47,82,20)(25,94,72,57)(26,95,61,58)(27,96,62,59)(28,85,63,60)(29,86,64,49)(30,87,65,50)(31,88,66,51)(32,89,67,52)(33,90,68,53)(34,91,69,54)(35,92,70,55)(36,93,71,56), (1,95,83,58)(2,96,84,59)(3,85,73,60)(4,86,74,49)(5,87,75,50)(6,88,76,51)(7,89,77,52)(8,90,78,53)(9,91,79,54)(10,92,80,55)(11,93,81,56)(12,94,82,57)(13,65,40,30)(14,66,41,31)(15,67,42,32)(16,68,43,33)(17,69,44,34)(18,70,45,35)(19,71,46,36)(20,72,47,25)(21,61,48,26)(22,62,37,27)(23,63,38,28)(24,64,39,29) );
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)], [(1,48,83,21),(2,37,84,22),(3,38,73,23),(4,39,74,24),(5,40,75,13),(6,41,76,14),(7,42,77,15),(8,43,78,16),(9,44,79,17),(10,45,80,18),(11,46,81,19),(12,47,82,20),(25,94,72,57),(26,95,61,58),(27,96,62,59),(28,85,63,60),(29,86,64,49),(30,87,65,50),(31,88,66,51),(32,89,67,52),(33,90,68,53),(34,91,69,54),(35,92,70,55),(36,93,71,56)], [(1,95,83,58),(2,96,84,59),(3,85,73,60),(4,86,74,49),(5,87,75,50),(6,88,76,51),(7,89,77,52),(8,90,78,53),(9,91,79,54),(10,92,80,55),(11,93,81,56),(12,94,82,57),(13,65,40,30),(14,66,41,31),(15,67,42,32),(16,68,43,33),(17,69,44,34),(18,70,45,35),(19,71,46,36),(20,72,47,25),(21,61,48,26),(22,62,37,27),(23,63,38,28),(24,64,39,29)]])
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