metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.202D6, D6.2M4(2), C12.10M4(2), C4⋊C8⋊11S3, D6⋊C8.11C2, C24⋊C4⋊21C2, C12⋊C8⋊13C2, (C2×C8).181D6, (S3×C42).3C2, C4.11(C8⋊S3), (C4×C12).61C22, C3⋊3(C42.6C4), (C4×Dic3).19C4, C6.27(C2×M4(2)), C2.16(S3×M4(2)), C12.305(C4○D4), (C2×C12).832C23, (C2×C24).254C22, C4.54(Q8⋊3S3), C4.131(D4⋊2S3), C6.32(C42⋊C2), (C4×Dic3).276C22, (C3×C4⋊C8)⋊21C2, (S3×C2×C4).19C4, C2.12(C2×C8⋊S3), (C2×C12).70(C2×C4), (C2×C4).146(C4×S3), C22.110(S3×C2×C4), C2.9(C4⋊C4⋊7S3), (C2×C3⋊C8).194C22, (S3×C2×C4).278C22, (C2×C6).87(C22×C4), (C22×S3).58(C2×C4), (C2×C4).774(C22×S3), (C2×Dic3).89(C2×C4), SmallGroup(192,394)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.202D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a2b-1, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2c5 >
Subgroups: 248 in 110 conjugacy classes, 53 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C12, C12, D6, D6, C2×C6, C42, C42, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, C42.6C4, C12⋊C8, C24⋊C4, D6⋊C8, C3×C4⋊C8, S3×C42, C42.202D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C2×M4(2), C8⋊S3, S3×C2×C4, D4⋊2S3, Q8⋊3S3, C42.6C4, C4⋊C4⋊7S3, C2×C8⋊S3, S3×M4(2), C42.202D6
►On 96 points
(1 39 59 95)(2 96 60 40)(3 41 61 73)(4 74 62 42)(5 43 63 75)(6 76 64 44)(7 45 65 77)(8 78 66 46)(9 47 67 79)(10 80 68 48)(11 25 69 81)(12 82 70 26)(13 27 71 83)(14 84 72 28)(15 29 49 85)(16 86 50 30)(17 31 51 87)(18 88 52 32)(19 33 53 89)(20 90 54 34)(21 35 55 91)(22 92 56 36)(23 37 57 93)(24 94 58 38)
(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 65 58 13 6 53 70)(2 57 66 5 14 69 54 17)(3 4 67 68 15 16 55 56)(7 24 71 64 19 12 59 52)(8 63 72 11 20 51 60 23)(9 10 49 50 21 22 61 62)(25 34 87 96 37 46 75 84)(26 95 88 45 38 83 76 33)(27 44 89 82 39 32 77 94)(28 81 90 31 40 93 78 43)(29 30 91 92 41 42 79 80)(35 36 73 74 47 48 85 86)
G:=sub<Sym(96)| (1,39,59,95)(2,96,60,40)(3,41,61,73)(4,74,62,42)(5,43,63,75)(6,76,64,44)(7,45,65,77)(8,78,66,46)(9,47,67,79)(10,80,68,48)(11,25,69,81)(12,82,70,26)(13,27,71,83)(14,84,72,28)(15,29,49,85)(16,86,50,30)(17,31,51,87)(18,88,52,32)(19,33,53,89)(20,90,54,34)(21,35,55,91)(22,92,56,36)(23,37,57,93)(24,94,58,38), (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), (1,18,65,58,13,6,53,70)(2,57,66,5,14,69,54,17)(3,4,67,68,15,16,55,56)(7,24,71,64,19,12,59,52)(8,63,72,11,20,51,60,23)(9,10,49,50,21,22,61,62)(25,34,87,96,37,46,75,84)(26,95,88,45,38,83,76,33)(27,44,89,82,39,32,77,94)(28,81,90,31,40,93,78,43)(29,30,91,92,41,42,79,80)(35,36,73,74,47,48,85,86)>;
G:=Group( (1,39,59,95)(2,96,60,40)(3,41,61,73)(4,74,62,42)(5,43,63,75)(6,76,64,44)(7,45,65,77)(8,78,66,46)(9,47,67,79)(10,80,68,48)(11,25,69,81)(12,82,70,26)(13,27,71,83)(14,84,72,28)(15,29,49,85)(16,86,50,30)(17,31,51,87)(18,88,52,32)(19,33,53,89)(20,90,54,34)(21,35,55,91)(22,92,56,36)(23,37,57,93)(24,94,58,38), (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), (1,18,65,58,13,6,53,70)(2,57,66,5,14,69,54,17)(3,4,67,68,15,16,55,56)(7,24,71,64,19,12,59,52)(8,63,72,11,20,51,60,23)(9,10,49,50,21,22,61,62)(25,34,87,96,37,46,75,84)(26,95,88,45,38,83,76,33)(27,44,89,82,39,32,77,94)(28,81,90,31,40,93,78,43)(29,30,91,92,41,42,79,80)(35,36,73,74,47,48,85,86) );
G=PermutationGroup([[(1,39,59,95),(2,96,60,40),(3,41,61,73),(4,74,62,42),(5,43,63,75),(6,76,64,44),(7,45,65,77),(8,78,66,46),(9,47,67,79),(10,80,68,48),(11,25,69,81),(12,82,70,26),(13,27,71,83),(14,84,72,28),(15,29,49,85),(16,86,50,30),(17,31,51,87),(18,88,52,32),(19,33,53,89),(20,90,54,34),(21,35,55,91),(22,92,56,36),(23,37,57,93),(24,94,58,38)], [(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)], [(1,18,65,58,13,6,53,70),(2,57,66,5,14,69,54,17),(3,4,67,68,15,16,55,56),(7,24,71,64,19,12,59,52),(8,63,72,11,20,51,60,23),(9,10,49,50,21,22,61,62),(25,34,87,96,37,46,75,84),(26,95,88,45,38,83,76,33),(27,44,89,82,39,32,77,94),(28,81,90,31,40,93,78,43),(29,30,91,92,41,42,79,80),(35,36,73,74,47,48,85,86)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4○D4 | M4(2) | C4×S3 | C8⋊S3 | D4⋊2S3 | Q8⋊3S3 | S3×M4(2) |
| kernel | C42.202D6 | C12⋊C8 | C24⋊C4 | D6⋊C8 | C3×C4⋊C8 | S3×C42 | C4×Dic3 | S3×C2×C4 | C4⋊C8 | C42 | C2×C8 | C12 | C12 | D6 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C42.202D6 ►in GL4(𝔽73) generated by
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 3 | 70 |
| 0 | 0 | 3 | 6 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 70 | 3 |
| 0 | 0 | 6 | 3 |
G:=sub<GL(4,GF(73))| [46,0,0,0,0,27,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,46,0,0,0,0,46],[0,72,0,0,1,0,0,0,0,0,3,3,0,0,70,6],[0,72,0,0,72,0,0,0,0,0,70,6,0,0,3,3] >;
C42.202D6 in GAP, Magma, Sage, TeX
C_4^2._{202}D_6 % in TeX
G:=Group("C4^2.202D6"); // GroupNames label
G:=SmallGroup(192,394);
// by ID
G=gap.SmallGroup(192,394);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,1094,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^-1,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^5>;
// generators/relations