metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.36D6, Dic6.18D4, C4⋊C8⋊6S3, (C2×C8).131D6, C4.132(S3×D4), (C2×C4).39D12, (C4×Dic6)⋊17C2, (C2×Dic12)⋊7C2, C6.13(C4○D8), C12.341(C2×D4), (C2×C12).245D4, C3⋊3(Q8.D4), C2.D24.2C2, C2.15(C4○D24), C6.40(C4⋊D4), C42⋊7S3.5C2, (C4×C12).71C22, (C2×C24).24C22, C2.Dic12⋊13C2, C12.330(C4○D4), C2.13(C12⋊D4), (C2×C12).755C23, C4.46(Q8⋊3S3), C2.18(C8.D6), (C2×D12).16C22, C22.118(C2×D12), C6.15(C8.C22), C4⋊Dic3.275C22, (C2×Dic6).214C22, (C3×C4⋊C8)⋊8C2, (C2×C24⋊C2).6C2, (C2×C6).138(C2×D4), (C2×C4).700(C22×S3), SmallGroup(192,404)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.36D6
G = < a,b,c,d | a4=b4=1, c6=dbd-1=b-1, d2=b2, ab=ba, cac-1=a-1b2, ad=da, bc=cb, dcd-1=b-1c5 >
Subgroups: 360 in 112 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C24, Dic6, Dic6, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C24⋊C2, Dic12, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, C2×D12, Q8.D4, C2.Dic12, C2.D24, C3×C4⋊C8, C4×Dic6, C42⋊7S3, C2×C24⋊C2, C2×Dic12, C42.36D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C4○D8, C8.C22, C2×D12, S3×D4, Q8⋊3S3, Q8.D4, C12⋊D4, C4○D24, C8.D6, C42.36D6
►On 96 points
(1 33 69 91)(2 80 70 46)(3 35 71 93)(4 82 72 48)(5 37 49 95)(6 84 50 26)(7 39 51 73)(8 86 52 28)(9 41 53 75)(10 88 54 30)(11 43 55 77)(12 90 56 32)(13 45 57 79)(14 92 58 34)(15 47 59 81)(16 94 60 36)(17 25 61 83)(18 96 62 38)(19 27 63 85)(20 74 64 40)(21 29 65 87)(22 76 66 42)(23 31 67 89)(24 78 68 44)
(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 80 37 92)(26 91 38 79)(27 78 39 90)(28 89 40 77)(29 76 41 88)(30 87 42 75)(31 74 43 86)(32 85 44 73)(33 96 45 84)(34 83 46 95)(35 94 47 82)(36 81 48 93)(49 58 61 70)(50 69 62 57)(51 56 63 68)(52 67 64 55)(53 54 65 66)(59 72 71 60)
G:=sub<Sym(96)| (1,33,69,91)(2,80,70,46)(3,35,71,93)(4,82,72,48)(5,37,49,95)(6,84,50,26)(7,39,51,73)(8,86,52,28)(9,41,53,75)(10,88,54,30)(11,43,55,77)(12,90,56,32)(13,45,57,79)(14,92,58,34)(15,47,59,81)(16,94,60,36)(17,25,61,83)(18,96,62,38)(19,27,63,85)(20,74,64,40)(21,29,65,87)(22,76,66,42)(23,31,67,89)(24,78,68,44), (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)(25,80,37,92)(26,91,38,79)(27,78,39,90)(28,89,40,77)(29,76,41,88)(30,87,42,75)(31,74,43,86)(32,85,44,73)(33,96,45,84)(34,83,46,95)(35,94,47,82)(36,81,48,93)(49,58,61,70)(50,69,62,57)(51,56,63,68)(52,67,64,55)(53,54,65,66)(59,72,71,60)>;
G:=Group( (1,33,69,91)(2,80,70,46)(3,35,71,93)(4,82,72,48)(5,37,49,95)(6,84,50,26)(7,39,51,73)(8,86,52,28)(9,41,53,75)(10,88,54,30)(11,43,55,77)(12,90,56,32)(13,45,57,79)(14,92,58,34)(15,47,59,81)(16,94,60,36)(17,25,61,83)(18,96,62,38)(19,27,63,85)(20,74,64,40)(21,29,65,87)(22,76,66,42)(23,31,67,89)(24,78,68,44), (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)(25,80,37,92)(26,91,38,79)(27,78,39,90)(28,89,40,77)(29,76,41,88)(30,87,42,75)(31,74,43,86)(32,85,44,73)(33,96,45,84)(34,83,46,95)(35,94,47,82)(36,81,48,93)(49,58,61,70)(50,69,62,57)(51,56,63,68)(52,67,64,55)(53,54,65,66)(59,72,71,60) );
G=PermutationGroup([[(1,33,69,91),(2,80,70,46),(3,35,71,93),(4,82,72,48),(5,37,49,95),(6,84,50,26),(7,39,51,73),(8,86,52,28),(9,41,53,75),(10,88,54,30),(11,43,55,77),(12,90,56,32),(13,45,57,79),(14,92,58,34),(15,47,59,81),(16,94,60,36),(17,25,61,83),(18,96,62,38),(19,27,63,85),(20,74,64,40),(21,29,65,87),(22,76,66,42),(23,31,67,89),(24,78,68,44)], [(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),(25,80,37,92),(26,91,38,79),(27,78,39,90),(28,89,40,77),(29,76,41,88),(30,87,42,75),(31,74,43,86),(32,85,44,73),(33,96,45,84),(34,83,46,95),(35,94,47,82),(36,81,48,93),(49,58,61,70),(50,69,62,57),(51,56,63,68),(52,67,64,55),(53,54,65,66),(59,72,71,60)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D12 | C4○D8 | C4○D24 | C8.C22 | S3×D4 | Q8⋊3S3 | C8.D6 |
| kernel | C42.36D6 | C2.Dic12 | C2.D24 | C3×C4⋊C8 | C4×Dic6 | C42⋊7S3 | C2×C24⋊C2 | C2×Dic12 | C4⋊C8 | Dic6 | C2×C12 | C42 | C2×C8 | C12 | C2×C4 | C6 | C2 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of C42.36D6 ►in GL4(𝔽73) generated by
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 66 | 14 | 0 | 0 |
| 59 | 7 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 11 | 37 | 0 | 0 |
| 36 | 48 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 25 | 37 | 0 | 0 |
| 62 | 48 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [46,0,0,0,0,46,0,0,0,0,0,46,0,0,46,0],[66,59,0,0,14,7,0,0,0,0,72,0,0,0,0,72],[11,36,0,0,37,48,0,0,0,0,0,72,0,0,1,0],[25,62,0,0,37,48,0,0,0,0,0,72,0,0,72,0] >;
C42.36D6 in GAP, Magma, Sage, TeX
C_4^2._{36}D_6 % in TeX
G:=Group("C4^2.36D6"); // GroupNames label
G:=SmallGroup(192,404);
// by ID
G=gap.SmallGroup(192,404);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,219,58,1123,136,6278]);
// 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=a^-1*b^2,a*d=d*a,b*c=c*b,d*c*d^-1=b^-1*c^5>;
// generators/relations