metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.200D6, Dic3.6M4(2), C4⋊C8⋊17S3, (C4×S3)⋊2C8, C4.14(S3×C8), D6.4(C2×C8), C12.11(C2×C8), D6⋊C8.10C2, C12⋊C8⋊12C2, (C2×C8).215D6, (C8×Dic3)⋊23C2, (S3×C42).2C2, C6.10(C22×C8), Dic3.9(C2×C8), (C4×Dic3).6C4, C2.6(S3×M4(2)), (C4×C12).59C22, C6.26(C2×M4(2)), C12.304(C4○D4), (C2×C12).830C23, (C2×C24).252C22, C3⋊3(C42.12C4), C4.52(Q8⋊3S3), C4.130(D4⋊2S3), C6.31(C42⋊C2), (C4×Dic3).301C22, (S3×C2×C4).6C4, C2.12(S3×C2×C8), (C3×C4⋊C8)⋊19C2, C22.47(S3×C2×C4), (C2×C4).145(C4×S3), (C2×C12).69(C2×C4), C2.3(C4⋊C4⋊7S3), (C2×C3⋊C8).305C22, (S3×C2×C4).276C22, (C2×C6).85(C22×C4), (C22×S3).57(C2×C4), (C2×C4).772(C22×S3), (C2×Dic3).88(C2×C4), SmallGroup(192,392)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.200D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a2b, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=a2b2c5 >
Subgroups: 248 in 118 conjugacy classes, 61 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, C12, D6, D6, C2×C6, C42, C42, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C4×C8, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, C42.12C4, C12⋊C8, C8×Dic3, D6⋊C8, C3×C4⋊C8, S3×C42, C42.200D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, M4(2), C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C22×C8, C2×M4(2), S3×C8, S3×C2×C4, D4⋊2S3, Q8⋊3S3, C42.12C4, C4⋊C4⋊7S3, S3×C2×C8, S3×M4(2), C42.200D6
►On 96 points
(1 69 37 94)(2 95 38 70)(3 71 39 96)(4 73 40 72)(5 49 41 74)(6 75 42 50)(7 51 43 76)(8 77 44 52)(9 53 45 78)(10 79 46 54)(11 55 47 80)(12 81 48 56)(13 57 25 82)(14 83 26 58)(15 59 27 84)(16 85 28 60)(17 61 29 86)(18 87 30 62)(19 63 31 88)(20 89 32 64)(21 65 33 90)(22 91 34 66)(23 67 35 92)(24 93 36 68)
(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 31 48 13 6 43 36)(2 47 32 5 14 35 44 17)(3 4 33 34 15 16 45 46)(7 24 37 30 19 12 25 42)(8 29 38 11 20 41 26 23)(9 10 39 40 21 22 27 28)(49 58 92 77 61 70 80 89)(50 76 93 69 62 88 81 57)(51 68 94 87 63 56 82 75)(52 86 95 55 64 74 83 67)(53 54 96 73 65 66 84 85)(59 60 78 79 71 72 90 91)
G:=sub<Sym(96)| (1,69,37,94)(2,95,38,70)(3,71,39,96)(4,73,40,72)(5,49,41,74)(6,75,42,50)(7,51,43,76)(8,77,44,52)(9,53,45,78)(10,79,46,54)(11,55,47,80)(12,81,48,56)(13,57,25,82)(14,83,26,58)(15,59,27,84)(16,85,28,60)(17,61,29,86)(18,87,30,62)(19,63,31,88)(20,89,32,64)(21,65,33,90)(22,91,34,66)(23,67,35,92)(24,93,36,68), (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), (1,18,31,48,13,6,43,36)(2,47,32,5,14,35,44,17)(3,4,33,34,15,16,45,46)(7,24,37,30,19,12,25,42)(8,29,38,11,20,41,26,23)(9,10,39,40,21,22,27,28)(49,58,92,77,61,70,80,89)(50,76,93,69,62,88,81,57)(51,68,94,87,63,56,82,75)(52,86,95,55,64,74,83,67)(53,54,96,73,65,66,84,85)(59,60,78,79,71,72,90,91)>;
G:=Group( (1,69,37,94)(2,95,38,70)(3,71,39,96)(4,73,40,72)(5,49,41,74)(6,75,42,50)(7,51,43,76)(8,77,44,52)(9,53,45,78)(10,79,46,54)(11,55,47,80)(12,81,48,56)(13,57,25,82)(14,83,26,58)(15,59,27,84)(16,85,28,60)(17,61,29,86)(18,87,30,62)(19,63,31,88)(20,89,32,64)(21,65,33,90)(22,91,34,66)(23,67,35,92)(24,93,36,68), (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), (1,18,31,48,13,6,43,36)(2,47,32,5,14,35,44,17)(3,4,33,34,15,16,45,46)(7,24,37,30,19,12,25,42)(8,29,38,11,20,41,26,23)(9,10,39,40,21,22,27,28)(49,58,92,77,61,70,80,89)(50,76,93,69,62,88,81,57)(51,68,94,87,63,56,82,75)(52,86,95,55,64,74,83,67)(53,54,96,73,65,66,84,85)(59,60,78,79,71,72,90,91) );
G=PermutationGroup([[(1,69,37,94),(2,95,38,70),(3,71,39,96),(4,73,40,72),(5,49,41,74),(6,75,42,50),(7,51,43,76),(8,77,44,52),(9,53,45,78),(10,79,46,54),(11,55,47,80),(12,81,48,56),(13,57,25,82),(14,83,26,58),(15,59,27,84),(16,85,28,60),(17,61,29,86),(18,87,30,62),(19,63,31,88),(20,89,32,64),(21,65,33,90),(22,91,34,66),(23,67,35,92),(24,93,36,68)], [(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)], [(1,18,31,48,13,6,43,36),(2,47,32,5,14,35,44,17),(3,4,33,34,15,16,45,46),(7,24,37,30,19,12,25,42),(8,29,38,11,20,41,26,23),(9,10,39,40,21,22,27,28),(49,58,92,77,61,70,80,89),(50,76,93,69,62,88,81,57),(51,68,94,87,63,56,82,75),(52,86,95,55,64,74,83,67),(53,54,96,73,65,66,84,85),(59,60,78,79,71,72,90,91)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 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 | 4 | 4 | 6 | 6 | 6 | 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 | 3 | ··· | 3 | 6 | 6 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | D6 | M4(2) | C4○D4 | C4×S3 | S3×C8 | D4⋊2S3 | Q8⋊3S3 | S3×M4(2) |
| kernel | C42.200D6 | C12⋊C8 | C8×Dic3 | D6⋊C8 | C3×C4⋊C8 | S3×C42 | C4×Dic3 | S3×C2×C4 | C4×S3 | C4⋊C8 | C42 | C2×C8 | Dic3 | C12 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 16 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C42.200D6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 46 | 27 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 51 | 51 | 0 | 0 |
| 22 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
| 51 | 51 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,46,46,0,0,0,27],[46,0,0,0,0,46,0,0,0,0,72,0,0,0,0,72],[51,22,0,0,51,0,0,0,0,0,1,1,0,0,71,72],[51,0,0,0,51,22,0,0,0,0,72,0,0,0,2,1] >;
C42.200D6 in GAP, Magma, Sage, TeX
C_4^2._{200}D_6 % in TeX
G:=Group("C4^2.200D6"); // GroupNames label
G:=SmallGroup(192,392);
// by ID
G=gap.SmallGroup(192,392);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,422,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^-1,d^2=a^2*b,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*b^2*c^5>;
// generators/relations