metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D24.3C4, C8.23D12, C24.39D4, C12.56D8, Dic12.3C4, C8.14(C4×S3), C8.C4⋊2S3, C24.12(C2×C4), C4.6(D6⋊C4), C4○D24.3C2, (C2×C12).97D4, (C2×C8).249D6, (C2×C6).6SD16, C4.29(D4⋊S3), C3⋊1(D8.C4), C6.9(D4⋊C4), C12.6(C22⋊C4), (C2×C24).37C22, C2.11(C6.D8), C22.1(Q8⋊2S3), (C2×C3⋊C16)⋊1C2, (C3×C8.C4)⋊1C2, (C2×C4).118(C3⋊D4), SmallGroup(192,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic12.C4
G = < a,b,c | a24=1, b2=c4=a12, bab-1=a-1, cac-1=a7, cbc-1=a9b >
Smallest permutation representation of Dic12.C4
►On 96 points
Generators in S96
(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 30 13 42)(2 29 14 41)(3 28 15 40)(4 27 16 39)(5 26 17 38)(6 25 18 37)(7 48 19 36)(8 47 20 35)(9 46 21 34)(10 45 22 33)(11 44 23 32)(12 43 24 31)(49 93 61 81)(50 92 62 80)(51 91 63 79)(52 90 64 78)(53 89 65 77)(54 88 66 76)(55 87 67 75)(56 86 68 74)(57 85 69 73)(58 84 70 96)(59 83 71 95)(60 82 72 94)
(1 76 30 69 13 88 42 57)(2 83 31 52 14 95 43 64)(3 90 32 59 15 78 44 71)(4 73 33 66 16 85 45 54)(5 80 34 49 17 92 46 61)(6 87 35 56 18 75 47 68)(7 94 36 63 19 82 48 51)(8 77 37 70 20 89 25 58)(9 84 38 53 21 96 26 65)(10 91 39 60 22 79 27 72)(11 74 40 67 23 86 28 55)(12 81 41 50 24 93 29 62)
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), (1,30,13,42)(2,29,14,41)(3,28,15,40)(4,27,16,39)(5,26,17,38)(6,25,18,37)(7,48,19,36)(8,47,20,35)(9,46,21,34)(10,45,22,33)(11,44,23,32)(12,43,24,31)(49,93,61,81)(50,92,62,80)(51,91,63,79)(52,90,64,78)(53,89,65,77)(54,88,66,76)(55,87,67,75)(56,86,68,74)(57,85,69,73)(58,84,70,96)(59,83,71,95)(60,82,72,94), (1,76,30,69,13,88,42,57)(2,83,31,52,14,95,43,64)(3,90,32,59,15,78,44,71)(4,73,33,66,16,85,45,54)(5,80,34,49,17,92,46,61)(6,87,35,56,18,75,47,68)(7,94,36,63,19,82,48,51)(8,77,37,70,20,89,25,58)(9,84,38,53,21,96,26,65)(10,91,39,60,22,79,27,72)(11,74,40,67,23,86,28,55)(12,81,41,50,24,93,29,62)>;
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), (1,30,13,42)(2,29,14,41)(3,28,15,40)(4,27,16,39)(5,26,17,38)(6,25,18,37)(7,48,19,36)(8,47,20,35)(9,46,21,34)(10,45,22,33)(11,44,23,32)(12,43,24,31)(49,93,61,81)(50,92,62,80)(51,91,63,79)(52,90,64,78)(53,89,65,77)(54,88,66,76)(55,87,67,75)(56,86,68,74)(57,85,69,73)(58,84,70,96)(59,83,71,95)(60,82,72,94), (1,76,30,69,13,88,42,57)(2,83,31,52,14,95,43,64)(3,90,32,59,15,78,44,71)(4,73,33,66,16,85,45,54)(5,80,34,49,17,92,46,61)(6,87,35,56,18,75,47,68)(7,94,36,63,19,82,48,51)(8,77,37,70,20,89,25,58)(9,84,38,53,21,96,26,65)(10,91,39,60,22,79,27,72)(11,74,40,67,23,86,28,55)(12,81,41,50,24,93,29,62) );
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)], [(1,30,13,42),(2,29,14,41),(3,28,15,40),(4,27,16,39),(5,26,17,38),(6,25,18,37),(7,48,19,36),(8,47,20,35),(9,46,21,34),(10,45,22,33),(11,44,23,32),(12,43,24,31),(49,93,61,81),(50,92,62,80),(51,91,63,79),(52,90,64,78),(53,89,65,77),(54,88,66,76),(55,87,67,75),(56,86,68,74),(57,85,69,73),(58,84,70,96),(59,83,71,95),(60,82,72,94)], [(1,76,30,69,13,88,42,57),(2,83,31,52,14,95,43,64),(3,90,32,59,15,78,44,71),(4,73,33,66,16,85,45,54),(5,80,34,49,17,92,46,61),(6,87,35,56,18,75,47,68),(7,94,36,63,19,82,48,51),(8,77,37,70,20,89,25,58),(9,84,38,53,21,96,26,65),(10,91,39,60,22,79,27,72),(11,74,40,67,23,86,28,55),(12,81,41,50,24,93,29,62)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 16A | ··· | 16H | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 24 | 2 | 1 | 1 | 2 | 24 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | D8.C4 | D4⋊S3 | Q8⋊2S3 | Dic12.C4 |
| kernel | Dic12.C4 | C2×C3⋊C16 | C3×C8.C4 | C4○D24 | D24 | Dic12 | C8.C4 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C3 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of Dic12.C4 ►in GL4(𝔽97) generated by
| 0 | 1 | 0 | 0 |
| 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 90 | 14 |
| 96 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 75 | 0 |
| 0 | 0 | 75 | 22 |
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 61 | 78 |
| 0 | 0 | 3 | 36 |
G:=sub<GL(4,GF(97))| [0,96,0,0,1,96,0,0,0,0,0,90,0,0,14,14],[96,1,0,0,0,1,0,0,0,0,75,75,0,0,0,22],[22,0,0,0,0,22,0,0,0,0,61,3,0,0,78,36] >;
Dic12.C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{12}.C_4 % in TeX
G:=Group("Dic12.C4"); // GroupNames label
G:=SmallGroup(192,56);
// by ID
G=gap.SmallGroup(192,56);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,184,675,346,192,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^2=c^4=a^12,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations
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