metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.187D6, C12⋊C8⋊7C2, C4⋊C4.6Dic3, C6.37(C8○D4), C42⋊C2.8S3, (C4×C12).19C22, (C22×C4).125D6, C22⋊C4.3Dic3, C12.250(C4○D4), C4.134(C4○D12), C42.S3⋊22C2, (C2×C12).847C23, C2.5(D4.Dic3), C23.13(C2×Dic3), C6.46(C42⋊C2), C12.55D4.17C2, C3⋊5(C42.7C22), (C22×C12).373C22, C22.44(C22×Dic3), C2.10(C23.26D6), (C4×C3⋊C8)⋊26C2, (C3×C4⋊C4).9C4, (C3×C22⋊C4).4C4, (C2×C12).161(C2×C4), (C2×C3⋊C8).314C22, (C22×C6).58(C2×C4), (C2×C4).19(C2×Dic3), (C2×C6).184(C22×C4), (C2×C4).789(C22×S3), (C3×C42⋊C2).9C2, SmallGroup(192,559)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.187D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2c-1 >
Subgroups: 168 in 96 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C2×C3⋊C8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C42.7C22, C4×C3⋊C8, C42.S3, C12⋊C8, C12.55D4, C3×C42⋊C2, C42.187D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, C2×Dic3, C22×S3, C42⋊C2, C8○D4, C4○D12, C22×Dic3, C42.7C22, C23.26D6, D4.Dic3, C42.187D6
►On 96 points
Generators in S96
(1 50 79 86)(2 55 80 83)(3 52 73 88)(4 49 74 85)(5 54 75 82)(6 51 76 87)(7 56 77 84)(8 53 78 81)(9 66 95 39)(10 71 96 36)(11 68 89 33)(12 65 90 38)(13 70 91 35)(14 67 92 40)(15 72 93 37)(16 69 94 34)(17 43 32 62)(18 48 25 59)(19 45 26 64)(20 42 27 61)(21 47 28 58)(22 44 29 63)(23 41 30 60)(24 46 31 57)
(1 73 5 77)(2 74 6 78)(3 75 7 79)(4 76 8 80)(9 89 13 93)(10 90 14 94)(11 91 15 95)(12 92 16 96)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)(33 70 37 66)(34 71 38 67)(35 72 39 68)(36 65 40 69)(41 62 45 58)(42 63 46 59)(43 64 47 60)(44 57 48 61)(49 87 53 83)(50 88 54 84)(51 81 55 85)(52 82 56 86)
(1 95 62)(2 44 96 80 63 10)(3 89 64)(4 46 90 74 57 12)(5 91 58)(6 48 92 76 59 14)(7 93 60)(8 42 94 78 61 16)(9 43 79)(11 45 73)(13 47 75)(15 41 77)(17 54 39 21 50 35)(18 71 51 29 40 83)(19 56 33 23 52 37)(20 65 53 31 34 85)(22 67 55 25 36 87)(24 69 49 27 38 81)(26 84 68 30 88 72)(28 86 70 32 82 66)
(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)
G:=sub<Sym(96)| (1,50,79,86)(2,55,80,83)(3,52,73,88)(4,49,74,85)(5,54,75,82)(6,51,76,87)(7,56,77,84)(8,53,78,81)(9,66,95,39)(10,71,96,36)(11,68,89,33)(12,65,90,38)(13,70,91,35)(14,67,92,40)(15,72,93,37)(16,69,94,34)(17,43,32,62)(18,48,25,59)(19,45,26,64)(20,42,27,61)(21,47,28,58)(22,44,29,63)(23,41,30,60)(24,46,31,57), (1,73,5,77)(2,74,6,78)(3,75,7,79)(4,76,8,80)(9,89,13,93)(10,90,14,94)(11,91,15,95)(12,92,16,96)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25)(33,70,37,66)(34,71,38,67)(35,72,39,68)(36,65,40,69)(41,62,45,58)(42,63,46,59)(43,64,47,60)(44,57,48,61)(49,87,53,83)(50,88,54,84)(51,81,55,85)(52,82,56,86), (1,95,62)(2,44,96,80,63,10)(3,89,64)(4,46,90,74,57,12)(5,91,58)(6,48,92,76,59,14)(7,93,60)(8,42,94,78,61,16)(9,43,79)(11,45,73)(13,47,75)(15,41,77)(17,54,39,21,50,35)(18,71,51,29,40,83)(19,56,33,23,52,37)(20,65,53,31,34,85)(22,67,55,25,36,87)(24,69,49,27,38,81)(26,84,68,30,88,72)(28,86,70,32,82,66), (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)>;
G:=Group( (1,50,79,86)(2,55,80,83)(3,52,73,88)(4,49,74,85)(5,54,75,82)(6,51,76,87)(7,56,77,84)(8,53,78,81)(9,66,95,39)(10,71,96,36)(11,68,89,33)(12,65,90,38)(13,70,91,35)(14,67,92,40)(15,72,93,37)(16,69,94,34)(17,43,32,62)(18,48,25,59)(19,45,26,64)(20,42,27,61)(21,47,28,58)(22,44,29,63)(23,41,30,60)(24,46,31,57), (1,73,5,77)(2,74,6,78)(3,75,7,79)(4,76,8,80)(9,89,13,93)(10,90,14,94)(11,91,15,95)(12,92,16,96)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25)(33,70,37,66)(34,71,38,67)(35,72,39,68)(36,65,40,69)(41,62,45,58)(42,63,46,59)(43,64,47,60)(44,57,48,61)(49,87,53,83)(50,88,54,84)(51,81,55,85)(52,82,56,86), (1,95,62)(2,44,96,80,63,10)(3,89,64)(4,46,90,74,57,12)(5,91,58)(6,48,92,76,59,14)(7,93,60)(8,42,94,78,61,16)(9,43,79)(11,45,73)(13,47,75)(15,41,77)(17,54,39,21,50,35)(18,71,51,29,40,83)(19,56,33,23,52,37)(20,65,53,31,34,85)(22,67,55,25,36,87)(24,69,49,27,38,81)(26,84,68,30,88,72)(28,86,70,32,82,66), (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) );
G=PermutationGroup([[(1,50,79,86),(2,55,80,83),(3,52,73,88),(4,49,74,85),(5,54,75,82),(6,51,76,87),(7,56,77,84),(8,53,78,81),(9,66,95,39),(10,71,96,36),(11,68,89,33),(12,65,90,38),(13,70,91,35),(14,67,92,40),(15,72,93,37),(16,69,94,34),(17,43,32,62),(18,48,25,59),(19,45,26,64),(20,42,27,61),(21,47,28,58),(22,44,29,63),(23,41,30,60),(24,46,31,57)], [(1,73,5,77),(2,74,6,78),(3,75,7,79),(4,76,8,80),(9,89,13,93),(10,90,14,94),(11,91,15,95),(12,92,16,96),(17,26,21,30),(18,27,22,31),(19,28,23,32),(20,29,24,25),(33,70,37,66),(34,71,38,67),(35,72,39,68),(36,65,40,69),(41,62,45,58),(42,63,46,59),(43,64,47,60),(44,57,48,61),(49,87,53,83),(50,88,54,84),(51,81,55,85),(52,82,56,86)], [(1,95,62),(2,44,96,80,63,10),(3,89,64),(4,46,90,74,57,12),(5,91,58),(6,48,92,76,59,14),(7,93,60),(8,42,94,78,61,16),(9,43,79),(11,45,73),(13,47,75),(15,41,77),(17,54,39,21,50,35),(18,71,51,29,40,83),(19,56,33,23,52,37),(20,65,53,31,34,85),(22,67,55,25,36,87),(24,69,49,27,38,81),(26,84,68,30,88,72),(28,86,70,32,82,66)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | Dic3 | Dic3 | D6 | C4○D4 | C8○D4 | C4○D12 | D4.Dic3 |
| kernel | C42.187D6 | C4×C3⋊C8 | C42.S3 | C12⋊C8 | C12.55D4 | C3×C42⋊C2 | C3×C22⋊C4 | C3×C4⋊C4 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 1 | 2 | 2 | 2 | 1 | 4 | 8 | 8 | 4 |
Matrix representation of C42.187D6 ►in GL4(𝔽73) generated by
| 27 | 71 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 27 | 72 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 67 | 9 |
| 22 | 20 | 0 | 0 |
| 10 | 51 | 0 | 0 |
| 0 | 0 | 33 | 31 |
| 0 | 0 | 19 | 40 |
G:=sub<GL(4,GF(73))| [27,0,0,0,71,46,0,0,0,0,46,0,0,0,0,46],[46,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[1,27,0,0,0,72,0,0,0,0,8,67,0,0,0,9],[22,10,0,0,20,51,0,0,0,0,33,19,0,0,31,40] >;
C42.187D6 in GAP, Magma, Sage, TeX
C_4^2._{187}D_6 % in TeX
G:=Group("C4^2.187D6"); // GroupNames label
G:=SmallGroup(192,559);
// by ID
G=gap.SmallGroup(192,559);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,422,387,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations