metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.162D6, C6.1012- 1+4, C4⋊C4.213D6, C42⋊2C2⋊5S3, C42⋊2S3⋊7C2, D6⋊Q8⋊44C2, (C4×Dic6)⋊15C2, C22⋊C4.80D6, D6.27(C4○D4), Dic3.Q8⋊38C2, (C4×C12).34C22, (C2×C6).252C24, C2.65(Q8○D12), C4.Dic6⋊40C2, C23.9D6.4C2, (C2×C12).604C23, D6⋊C4.140C22, Dic3⋊4D4.5C2, C23.8D6⋊46C2, C23.68(C22×S3), (C22×C6).66C23, Dic3.32(C4○D4), Dic3.D4⋊46C2, C23.16D6⋊22C2, C4⋊Dic3.247C22, C22.273(S3×C23), Dic3⋊C4.146C22, (C22×S3).226C23, (C2×Dic6).255C22, (C2×Dic3).130C23, (C4×Dic3).218C22, C6.D4.68C22, C3⋊11(C22.46C24), (C22×Dic3).152C22, (S3×C4⋊C4)⋊42C2, C4⋊C4⋊7S3⋊41C2, C2.99(S3×C4○D4), C6.210(C2×C4○D4), (C3×C42⋊2C2)⋊7C2, (S3×C2×C4).220C22, (C2×C4).88(C22×S3), (C3×C4⋊C4).204C22, (C2×C3⋊D4).72C22, (C3×C22⋊C4).77C22, SmallGroup(192,1267)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.162D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >
Subgroups: 480 in 214 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22.46C24, C4×Dic6, C42⋊2S3, C23.16D6, Dic3.D4, C23.8D6, Dic3⋊4D4, C23.9D6, Dic3.Q8, C4.Dic6, S3×C4⋊C4, C4⋊C4⋊7S3, D6⋊Q8, C3×C42⋊2C2, C42.162D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, S3×C23, C22.46C24, S3×C4○D4, Q8○D12, C42.162D6
►On 96 points
(1 23 43 73)(2 18 44 80)(3 13 45 75)(4 20 46 82)(5 15 47 77)(6 22 48 84)(7 17 37 79)(8 24 38 74)(9 19 39 81)(10 14 40 76)(11 21 41 83)(12 16 42 78)(25 51 92 66)(26 58 93 61)(27 53 94 68)(28 60 95 63)(29 55 96 70)(30 50 85 65)(31 57 86 72)(32 52 87 67)(33 59 88 62)(34 54 89 69)(35 49 90 64)(36 56 91 71)
(1 49 7 55)(2 71 8 65)(3 51 9 57)(4 61 10 67)(5 53 11 59)(6 63 12 69)(13 92 19 86)(14 32 20 26)(15 94 21 88)(16 34 22 28)(17 96 23 90)(18 36 24 30)(25 81 31 75)(27 83 33 77)(29 73 35 79)(37 70 43 64)(38 50 44 56)(39 72 45 66)(40 52 46 58)(41 62 47 68)(42 54 48 60)(74 85 80 91)(76 87 82 93)(78 89 84 95)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 14 19 20)(15 24 21 18)(16 17 22 23)(25 93 31 87)(26 86 32 92)(27 91 33 85)(28 96 34 90)(29 89 35 95)(30 94 36 88)(37 42 43 48)(38 47 44 41)(39 40 45 46)(49 69 55 63)(50 62 56 68)(51 67 57 61)(52 72 58 66)(53 65 59 71)(54 70 60 64)(73 78 79 84)(74 83 80 77)(75 76 81 82)
G:=sub<Sym(96)| (1,23,43,73)(2,18,44,80)(3,13,45,75)(4,20,46,82)(5,15,47,77)(6,22,48,84)(7,17,37,79)(8,24,38,74)(9,19,39,81)(10,14,40,76)(11,21,41,83)(12,16,42,78)(25,51,92,66)(26,58,93,61)(27,53,94,68)(28,60,95,63)(29,55,96,70)(30,50,85,65)(31,57,86,72)(32,52,87,67)(33,59,88,62)(34,54,89,69)(35,49,90,64)(36,56,91,71), (1,49,7,55)(2,71,8,65)(3,51,9,57)(4,61,10,67)(5,53,11,59)(6,63,12,69)(13,92,19,86)(14,32,20,26)(15,94,21,88)(16,34,22,28)(17,96,23,90)(18,36,24,30)(25,81,31,75)(27,83,33,77)(29,73,35,79)(37,70,43,64)(38,50,44,56)(39,72,45,66)(40,52,46,58)(41,62,47,68)(42,54,48,60)(74,85,80,91)(76,87,82,93)(78,89,84,95), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,14,19,20)(15,24,21,18)(16,17,22,23)(25,93,31,87)(26,86,32,92)(27,91,33,85)(28,96,34,90)(29,89,35,95)(30,94,36,88)(37,42,43,48)(38,47,44,41)(39,40,45,46)(49,69,55,63)(50,62,56,68)(51,67,57,61)(52,72,58,66)(53,65,59,71)(54,70,60,64)(73,78,79,84)(74,83,80,77)(75,76,81,82)>;
G:=Group( (1,23,43,73)(2,18,44,80)(3,13,45,75)(4,20,46,82)(5,15,47,77)(6,22,48,84)(7,17,37,79)(8,24,38,74)(9,19,39,81)(10,14,40,76)(11,21,41,83)(12,16,42,78)(25,51,92,66)(26,58,93,61)(27,53,94,68)(28,60,95,63)(29,55,96,70)(30,50,85,65)(31,57,86,72)(32,52,87,67)(33,59,88,62)(34,54,89,69)(35,49,90,64)(36,56,91,71), (1,49,7,55)(2,71,8,65)(3,51,9,57)(4,61,10,67)(5,53,11,59)(6,63,12,69)(13,92,19,86)(14,32,20,26)(15,94,21,88)(16,34,22,28)(17,96,23,90)(18,36,24,30)(25,81,31,75)(27,83,33,77)(29,73,35,79)(37,70,43,64)(38,50,44,56)(39,72,45,66)(40,52,46,58)(41,62,47,68)(42,54,48,60)(74,85,80,91)(76,87,82,93)(78,89,84,95), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,14,19,20)(15,24,21,18)(16,17,22,23)(25,93,31,87)(26,86,32,92)(27,91,33,85)(28,96,34,90)(29,89,35,95)(30,94,36,88)(37,42,43,48)(38,47,44,41)(39,40,45,46)(49,69,55,63)(50,62,56,68)(51,67,57,61)(52,72,58,66)(53,65,59,71)(54,70,60,64)(73,78,79,84)(74,83,80,77)(75,76,81,82) );
G=PermutationGroup([[(1,23,43,73),(2,18,44,80),(3,13,45,75),(4,20,46,82),(5,15,47,77),(6,22,48,84),(7,17,37,79),(8,24,38,74),(9,19,39,81),(10,14,40,76),(11,21,41,83),(12,16,42,78),(25,51,92,66),(26,58,93,61),(27,53,94,68),(28,60,95,63),(29,55,96,70),(30,50,85,65),(31,57,86,72),(32,52,87,67),(33,59,88,62),(34,54,89,69),(35,49,90,64),(36,56,91,71)], [(1,49,7,55),(2,71,8,65),(3,51,9,57),(4,61,10,67),(5,53,11,59),(6,63,12,69),(13,92,19,86),(14,32,20,26),(15,94,21,88),(16,34,22,28),(17,96,23,90),(18,36,24,30),(25,81,31,75),(27,83,33,77),(29,73,35,79),(37,70,43,64),(38,50,44,56),(39,72,45,66),(40,52,46,58),(41,62,47,68),(42,54,48,60),(74,85,80,91),(76,87,82,93),(78,89,84,95)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,14,19,20),(15,24,21,18),(16,17,22,23),(25,93,31,87),(26,86,32,92),(27,91,33,85),(28,96,34,90),(29,89,35,95),(30,94,36,88),(37,42,43,48),(38,47,44,41),(39,40,45,46),(49,69,55,63),(50,62,56,68),(51,67,57,61),(52,72,58,66),(53,65,59,71),(54,70,60,64),(73,78,79,84),(74,83,80,77),(75,76,81,82)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2- 1+4 | S3×C4○D4 | Q8○D12 |
| kernel | C42.162D6 | C4×Dic6 | C42⋊2S3 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | Dic3.Q8 | C4.Dic6 | S3×C4⋊C4 | C4⋊C4⋊7S3 | D6⋊Q8 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | D6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 4 | 2 |
Matrix representation of C42.162D6 ►in GL6(𝔽13)
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
C42.162D6 in GAP, Magma, Sage, TeX
C_4^2._{162}D_6 % in TeX
G:=Group("C4^2.162D6"); // GroupNames label
G:=SmallGroup(192,1267);
// by ID
G=gap.SmallGroup(192,1267);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations