metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.16D6, C8⋊1(C4×S3), C24⋊1(C2×C4), C8⋊C4⋊1S3, C24⋊C2⋊1C4, C6.11(C4×D4), (C2×C8).53D6, (C4×Dic6)⋊2C2, Dic6⋊9(C2×C4), (C4×D12).5C2, C24⋊1C4⋊16C2, C2.14(C4×D12), D12.14(C2×C4), (C2×C12).236D4, (C2×C4).114D12, C2.1(C8⋊D6), C6.2(C8⋊C22), C3⋊1(SD16⋊C4), (C2×C24).54C22, (C4×C12).14C22, C2.Dic12⋊37C2, C2.D24.15C2, C2.1(C8.D6), C22.30(C2×D12), C6.3(C8.C22), C12.222(C4○D4), C4.106(C4○D12), (C2×C12).731C23, C12.105(C22×C4), (C2×D12).189C22, C4⋊Dic3.265C22, (C2×Dic6).208C22, C4.63(S3×C2×C4), (C3×C8⋊C4)⋊2C2, (C2×C24⋊C2).1C2, (C2×C6).114(C2×D4), (C2×C4).675(C22×S3), SmallGroup(192,269)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.16D6
G = < a,b,c,d | a4=b4=1, c6=dbd-1=b-1, d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dcd-1=b-1c5 >
Subgroups: 360 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C24⋊C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, SD16⋊C4, C2.Dic12, C24⋊1C4, C2.D24, C3×C8⋊C4, C4×Dic6, C4×D12, C2×C24⋊C2, C42.16D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C8⋊C22, C8.C22, S3×C2×C4, C2×D12, C4○D12, SD16⋊C4, C4×D12, C8⋊D6, C8.D6, C42.16D6
►On 96 points
Generators in S96
(1 39 51 82)(2 28 52 95)(3 41 53 84)(4 30 54 73)(5 43 55 86)(6 32 56 75)(7 45 57 88)(8 34 58 77)(9 47 59 90)(10 36 60 79)(11 25 61 92)(12 38 62 81)(13 27 63 94)(14 40 64 83)(15 29 65 96)(16 42 66 85)(17 31 67 74)(18 44 68 87)(19 33 69 76)(20 46 70 89)(21 35 71 78)(22 48 72 91)(23 37 49 80)(24 26 50 93)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)
(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 18 13 6)(2 5 14 17)(3 16 15 4)(7 12 19 24)(8 23 20 11)(9 10 21 22)(25 46 37 34)(26 33 38 45)(27 44 39 32)(28 31 40 43)(29 42 41 30)(35 36 47 48)(49 70 61 58)(50 57 62 69)(51 68 63 56)(52 55 64 67)(53 66 65 54)(59 60 71 72)(73 96 85 84)(74 83 86 95)(75 94 87 82)(76 81 88 93)(77 92 89 80)(78 79 90 91)
G:=sub<Sym(96)| (1,39,51,82)(2,28,52,95)(3,41,53,84)(4,30,54,73)(5,43,55,86)(6,32,56,75)(7,45,57,88)(8,34,58,77)(9,47,59,90)(10,36,60,79)(11,25,61,92)(12,38,62,81)(13,27,63,94)(14,40,64,83)(15,29,65,96)(16,42,66,85)(17,31,67,74)(18,44,68,87)(19,33,69,76)(20,46,70,89)(21,35,71,78)(22,48,72,91)(23,37,49,80)(24,26,50,93), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84), (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), (1,18,13,6)(2,5,14,17)(3,16,15,4)(7,12,19,24)(8,23,20,11)(9,10,21,22)(25,46,37,34)(26,33,38,45)(27,44,39,32)(28,31,40,43)(29,42,41,30)(35,36,47,48)(49,70,61,58)(50,57,62,69)(51,68,63,56)(52,55,64,67)(53,66,65,54)(59,60,71,72)(73,96,85,84)(74,83,86,95)(75,94,87,82)(76,81,88,93)(77,92,89,80)(78,79,90,91)>;
G:=Group( (1,39,51,82)(2,28,52,95)(3,41,53,84)(4,30,54,73)(5,43,55,86)(6,32,56,75)(7,45,57,88)(8,34,58,77)(9,47,59,90)(10,36,60,79)(11,25,61,92)(12,38,62,81)(13,27,63,94)(14,40,64,83)(15,29,65,96)(16,42,66,85)(17,31,67,74)(18,44,68,87)(19,33,69,76)(20,46,70,89)(21,35,71,78)(22,48,72,91)(23,37,49,80)(24,26,50,93), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84), (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), (1,18,13,6)(2,5,14,17)(3,16,15,4)(7,12,19,24)(8,23,20,11)(9,10,21,22)(25,46,37,34)(26,33,38,45)(27,44,39,32)(28,31,40,43)(29,42,41,30)(35,36,47,48)(49,70,61,58)(50,57,62,69)(51,68,63,56)(52,55,64,67)(53,66,65,54)(59,60,71,72)(73,96,85,84)(74,83,86,95)(75,94,87,82)(76,81,88,93)(77,92,89,80)(78,79,90,91) );
G=PermutationGroup([[(1,39,51,82),(2,28,52,95),(3,41,53,84),(4,30,54,73),(5,43,55,86),(6,32,56,75),(7,45,57,88),(8,34,58,77),(9,47,59,90),(10,36,60,79),(11,25,61,92),(12,38,62,81),(13,27,63,94),(14,40,64,83),(15,29,65,96),(16,42,66,85),(17,31,67,74),(18,44,68,87),(19,33,69,76),(20,46,70,89),(21,35,71,78),(22,48,72,91),(23,37,49,80),(24,26,50,93)], [(1,19,13,7),(2,20,14,8),(3,21,15,9),(4,22,16,10),(5,23,17,11),(6,24,18,12),(25,43,37,31),(26,44,38,32),(27,45,39,33),(28,46,40,34),(29,47,41,35),(30,48,42,36),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60),(73,91,85,79),(74,92,86,80),(75,93,87,81),(76,94,88,82),(77,95,89,83),(78,96,90,84)], [(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)], [(1,18,13,6),(2,5,14,17),(3,16,15,4),(7,12,19,24),(8,23,20,11),(9,10,21,22),(25,46,37,34),(26,33,38,45),(27,44,39,32),(28,31,40,43),(29,42,41,30),(35,36,47,48),(49,70,61,58),(50,57,62,69),(51,68,63,56),(52,55,64,67),(53,66,65,54),(59,60,71,72),(73,96,85,84),(74,83,86,95),(75,94,87,82),(76,81,88,93),(77,92,89,80),(78,79,90,91)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | ··· | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 | C8⋊C22 | C8.C22 | C8⋊D6 | C8.D6 |
| kernel | C42.16D6 | C2.Dic12 | C24⋊1C4 | C2.D24 | C3×C8⋊C4 | C4×Dic6 | C4×D12 | C2×C24⋊C2 | C24⋊C2 | C8⋊C4 | C2×C12 | C42 | C2×C8 | C12 | C8 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C42.16D6 ►in GL8(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 21 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 23 | 52 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 37 | 4 | 59 |
| 0 | 0 | 0 | 0 | 16 | 45 | 63 | 54 |
| 0 | 0 | 0 | 0 | 4 | 59 | 21 | 36 |
| 0 | 0 | 0 | 0 | 63 | 54 | 57 | 28 |
| 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 36 | 54 | 14 |
| 0 | 0 | 0 | 0 | 43 | 28 | 37 | 19 |
| 0 | 0 | 0 | 0 | 54 | 14 | 28 | 37 |
| 0 | 0 | 0 | 0 | 37 | 19 | 30 | 45 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,21,23,0,0,0,0,0,0,3,52,0,0,0,0,0,0,0,0,21,23,0,0,0,0,0,0,3,52],[0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,64,23,0,0,0,0,0,0,25,9,0,0,0,0,0,0,0,0,52,16,4,63,0,0,0,0,37,45,59,54,0,0,0,0,4,63,21,57,0,0,0,0,59,54,36,28],[72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,9,47,0,0,0,0,0,0,48,64,0,0,0,0,0,0,0,0,45,43,54,37,0,0,0,0,36,28,14,19,0,0,0,0,54,37,28,30,0,0,0,0,14,19,37,45] >;
C42.16D6 in GAP, Magma, Sage, TeX
C_4^2._{16}D_6 % in TeX
G:=Group("C4^2.16D6"); // GroupNames label
G:=SmallGroup(192,269);
// by ID
G=gap.SmallGroup(192,269);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,387,58,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d*b*d^-1=b^-1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*c*d^-1=b^-1*c^5>;
// generators/relations