metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.16D8, C12.15SD16, C42.217D6, C6.56(C2×D8), C4.5(D4⋊S3), (C2×D4).53D6, C4⋊1D4.4S3, C3⋊3(C4.4D8), C12⋊2Q8⋊19C2, (C2×C12).146D4, C4.3(D4.S3), C6.57(C2×SD16), C12.74(C4○D4), D4⋊Dic3⋊20C2, (C6×D4).69C22, C4.22(D4⋊2S3), (C2×C12).388C23, (C4×C12).118C22, C6.43(C4.4D4), C4⋊Dic3.154C22, C2.10(C23.12D6), (C4×C3⋊C8)⋊14C2, C2.11(C2×D4⋊S3), (C3×C4⋊1D4).2C2, (C2×C6).519(C2×D4), C2.11(C2×D4.S3), (C2×C3⋊C8).256C22, (C2×C4).130(C3⋊D4), (C2×C4).486(C22×S3), C22.192(C2×C3⋊D4), SmallGroup(192,629)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.16D8
G = < a,b,c | a12=b8=1, c2=a6, bab-1=a5, cac-1=a-1, cbc-1=a6b-1 >
Subgroups: 336 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×C8, C2×D4, C2×D4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C4×C8, D4⋊C4, C4⋊1D4, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×Dic6, C6×D4, C6×D4, C4.4D8, C4×C3⋊C8, D4⋊Dic3, C12⋊2Q8, C3×C4⋊1D4, C12.16D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C2×D8, C2×SD16, D4⋊S3, D4.S3, D4⋊2S3, C2×C3⋊D4, C4.4D8, C2×D4⋊S3, C2×D4.S3, C23.12D6, C12.16D8
►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 91 13 76 37 63 50 30)(2 96 14 81 38 68 51 35)(3 89 15 74 39 61 52 28)(4 94 16 79 40 66 53 33)(5 87 17 84 41 71 54 26)(6 92 18 77 42 64 55 31)(7 85 19 82 43 69 56 36)(8 90 20 75 44 62 57 29)(9 95 21 80 45 67 58 34)(10 88 22 73 46 72 59 27)(11 93 23 78 47 65 60 32)(12 86 24 83 48 70 49 25)
(1 33 7 27)(2 32 8 26)(3 31 9 25)(4 30 10 36)(5 29 11 35)(6 28 12 34)(13 66 19 72)(14 65 20 71)(15 64 21 70)(16 63 22 69)(17 62 23 68)(18 61 24 67)(37 79 43 73)(38 78 44 84)(39 77 45 83)(40 76 46 82)(41 75 47 81)(42 74 48 80)(49 95 55 89)(50 94 56 88)(51 93 57 87)(52 92 58 86)(53 91 59 85)(54 90 60 96)
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), (1,91,13,76,37,63,50,30)(2,96,14,81,38,68,51,35)(3,89,15,74,39,61,52,28)(4,94,16,79,40,66,53,33)(5,87,17,84,41,71,54,26)(6,92,18,77,42,64,55,31)(7,85,19,82,43,69,56,36)(8,90,20,75,44,62,57,29)(9,95,21,80,45,67,58,34)(10,88,22,73,46,72,59,27)(11,93,23,78,47,65,60,32)(12,86,24,83,48,70,49,25), (1,33,7,27)(2,32,8,26)(3,31,9,25)(4,30,10,36)(5,29,11,35)(6,28,12,34)(13,66,19,72)(14,65,20,71)(15,64,21,70)(16,63,22,69)(17,62,23,68)(18,61,24,67)(37,79,43,73)(38,78,44,84)(39,77,45,83)(40,76,46,82)(41,75,47,81)(42,74,48,80)(49,95,55,89)(50,94,56,88)(51,93,57,87)(52,92,58,86)(53,91,59,85)(54,90,60,96)>;
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), (1,91,13,76,37,63,50,30)(2,96,14,81,38,68,51,35)(3,89,15,74,39,61,52,28)(4,94,16,79,40,66,53,33)(5,87,17,84,41,71,54,26)(6,92,18,77,42,64,55,31)(7,85,19,82,43,69,56,36)(8,90,20,75,44,62,57,29)(9,95,21,80,45,67,58,34)(10,88,22,73,46,72,59,27)(11,93,23,78,47,65,60,32)(12,86,24,83,48,70,49,25), (1,33,7,27)(2,32,8,26)(3,31,9,25)(4,30,10,36)(5,29,11,35)(6,28,12,34)(13,66,19,72)(14,65,20,71)(15,64,21,70)(16,63,22,69)(17,62,23,68)(18,61,24,67)(37,79,43,73)(38,78,44,84)(39,77,45,83)(40,76,46,82)(41,75,47,81)(42,74,48,80)(49,95,55,89)(50,94,56,88)(51,93,57,87)(52,92,58,86)(53,91,59,85)(54,90,60,96) );
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)], [(1,91,13,76,37,63,50,30),(2,96,14,81,38,68,51,35),(3,89,15,74,39,61,52,28),(4,94,16,79,40,66,53,33),(5,87,17,84,41,71,54,26),(6,92,18,77,42,64,55,31),(7,85,19,82,43,69,56,36),(8,90,20,75,44,62,57,29),(9,95,21,80,45,67,58,34),(10,88,22,73,46,72,59,27),(11,93,23,78,47,65,60,32),(12,86,24,83,48,70,49,25)], [(1,33,7,27),(2,32,8,26),(3,31,9,25),(4,30,10,36),(5,29,11,35),(6,28,12,34),(13,66,19,72),(14,65,20,71),(15,64,21,70),(16,63,22,69),(17,62,23,68),(18,61,24,67),(37,79,43,73),(38,78,44,84),(39,77,45,83),(40,76,46,82),(41,75,47,81),(42,74,48,80),(49,95,55,89),(50,94,56,88),(51,93,57,87),(52,92,58,86),(53,91,59,85),(54,90,60,96)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | ··· | 2 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D8 | SD16 | C4○D4 | C3⋊D4 | D4⋊S3 | D4.S3 | D4⋊2S3 |
| kernel | C12.16D8 | C4×C3⋊C8 | D4⋊Dic3 | C12⋊2Q8 | C3×C4⋊1D4 | C4⋊1D4 | C2×C12 | C42 | C2×D4 | C12 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C12.16D8 ►in GL6(𝔽73)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 57 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 27 | 54 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 57 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 37 | 0 |
G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,2,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,16,57,0,0,0,0,16,16,0,0,0,0,0,0,0,36,0,0,0,0,2,0],[27,0,0,0,0,0,54,46,0,0,0,0,0,0,57,57,0,0,0,0,57,16,0,0,0,0,0,0,0,37,0,0,0,0,2,0] >;
C12.16D8 in GAP, Magma, Sage, TeX
C_{12}._{16}D_8 % in TeX
G:=Group("C12.16D8"); // GroupNames label
G:=SmallGroup(192,629);
// by ID
G=gap.SmallGroup(192,629);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,590,135,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations