metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.163D6, C6.1402+ 1+4, (C4×D12)⋊16C2, C4⋊C4.214D6, Dic3⋊D4⋊45C2, C12⋊D4⋊37C2, C42⋊2S3⋊8C2, C42⋊2C2⋊6S3, C22⋊C4.81D6, Dic3⋊5D4⋊42C2, D6.33(C4○D4), D6.D4⋊42C2, C2.65(D4○D12), Dic3.Q8⋊39C2, (C2×C6).253C24, (C2×C12).96C23, (C4×C12).35C22, D6⋊C4.46C22, Dic3⋊4D4⋊37C2, C23.69(C22×S3), (C22×C6).67C23, Dic3.33(C4○D4), (C2×D12).169C22, C23.21D6⋊30C2, C4⋊Dic3.318C22, C22.274(S3×C23), Dic3⋊C4.147C22, (C22×S3).112C23, (C4×Dic3).152C22, (C2×Dic3).266C23, C3⋊11(C22.47C24), (C22×Dic3).153C22, (S3×C4⋊C4)⋊43C2, C4⋊C4⋊S3⋊43C2, C2.100(S3×C4○D4), C6.211(C2×C4○D4), (C3×C42⋊2C2)⋊8C2, (S3×C2×C4).135C22, (C2×C4).89(C22×S3), (C3×C4⋊C4).205C22, (C2×C3⋊D4).73C22, (C3×C22⋊C4).78C22, SmallGroup(192,1268)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.163D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 640 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, 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, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22.47C24, C42⋊2S3, C4×D12, Dic3⋊4D4, Dic3⋊D4, C23.21D6, Dic3.Q8, S3×C4⋊C4, Dic3⋊5D4, D6.D4, C12⋊D4, C4⋊C4⋊S3, C3×C42⋊2C2, C42.163D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, S3×C23, C22.47C24, S3×C4○D4, D4○D12, C42.163D6
►On 96 points
(1 83 7 77)(2 42 8 48)(3 73 9 79)(4 44 10 38)(5 75 11 81)(6 46 12 40)(13 31 19 25)(14 53 20 59)(15 33 21 27)(16 55 22 49)(17 35 23 29)(18 57 24 51)(26 65 32 71)(28 67 34 61)(30 69 36 63)(37 95 43 89)(39 85 45 91)(41 87 47 93)(50 68 56 62)(52 70 58 64)(54 72 60 66)(74 90 80 96)(76 92 82 86)(78 94 84 88)
(1 70 87 13)(2 65 88 20)(3 72 89 15)(4 67 90 22)(5 62 91 17)(6 69 92 24)(7 64 93 19)(8 71 94 14)(9 66 95 21)(10 61 96 16)(11 68 85 23)(12 63 86 18)(25 77 52 41)(26 84 53 48)(27 79 54 43)(28 74 55 38)(29 81 56 45)(30 76 57 40)(31 83 58 47)(32 78 59 42)(33 73 60 37)(34 80 49 44)(35 75 50 39)(36 82 51 46)
(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 21 7 15)(2 14 8 20)(3 19 9 13)(4 24 10 18)(5 17 11 23)(6 22 12 16)(25 37 31 43)(26 42 32 48)(27 47 33 41)(28 40 34 46)(29 45 35 39)(30 38 36 44)(49 82 55 76)(50 75 56 81)(51 80 57 74)(52 73 58 79)(53 78 59 84)(54 83 60 77)(61 92 67 86)(62 85 68 91)(63 90 69 96)(64 95 70 89)(65 88 71 94)(66 93 72 87)
G:=sub<Sym(96)| (1,83,7,77)(2,42,8,48)(3,73,9,79)(4,44,10,38)(5,75,11,81)(6,46,12,40)(13,31,19,25)(14,53,20,59)(15,33,21,27)(16,55,22,49)(17,35,23,29)(18,57,24,51)(26,65,32,71)(28,67,34,61)(30,69,36,63)(37,95,43,89)(39,85,45,91)(41,87,47,93)(50,68,56,62)(52,70,58,64)(54,72,60,66)(74,90,80,96)(76,92,82,86)(78,94,84,88), (1,70,87,13)(2,65,88,20)(3,72,89,15)(4,67,90,22)(5,62,91,17)(6,69,92,24)(7,64,93,19)(8,71,94,14)(9,66,95,21)(10,61,96,16)(11,68,85,23)(12,63,86,18)(25,77,52,41)(26,84,53,48)(27,79,54,43)(28,74,55,38)(29,81,56,45)(30,76,57,40)(31,83,58,47)(32,78,59,42)(33,73,60,37)(34,80,49,44)(35,75,50,39)(36,82,51,46), (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), (1,21,7,15)(2,14,8,20)(3,19,9,13)(4,24,10,18)(5,17,11,23)(6,22,12,16)(25,37,31,43)(26,42,32,48)(27,47,33,41)(28,40,34,46)(29,45,35,39)(30,38,36,44)(49,82,55,76)(50,75,56,81)(51,80,57,74)(52,73,58,79)(53,78,59,84)(54,83,60,77)(61,92,67,86)(62,85,68,91)(63,90,69,96)(64,95,70,89)(65,88,71,94)(66,93,72,87)>;
G:=Group( (1,83,7,77)(2,42,8,48)(3,73,9,79)(4,44,10,38)(5,75,11,81)(6,46,12,40)(13,31,19,25)(14,53,20,59)(15,33,21,27)(16,55,22,49)(17,35,23,29)(18,57,24,51)(26,65,32,71)(28,67,34,61)(30,69,36,63)(37,95,43,89)(39,85,45,91)(41,87,47,93)(50,68,56,62)(52,70,58,64)(54,72,60,66)(74,90,80,96)(76,92,82,86)(78,94,84,88), (1,70,87,13)(2,65,88,20)(3,72,89,15)(4,67,90,22)(5,62,91,17)(6,69,92,24)(7,64,93,19)(8,71,94,14)(9,66,95,21)(10,61,96,16)(11,68,85,23)(12,63,86,18)(25,77,52,41)(26,84,53,48)(27,79,54,43)(28,74,55,38)(29,81,56,45)(30,76,57,40)(31,83,58,47)(32,78,59,42)(33,73,60,37)(34,80,49,44)(35,75,50,39)(36,82,51,46), (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), (1,21,7,15)(2,14,8,20)(3,19,9,13)(4,24,10,18)(5,17,11,23)(6,22,12,16)(25,37,31,43)(26,42,32,48)(27,47,33,41)(28,40,34,46)(29,45,35,39)(30,38,36,44)(49,82,55,76)(50,75,56,81)(51,80,57,74)(52,73,58,79)(53,78,59,84)(54,83,60,77)(61,92,67,86)(62,85,68,91)(63,90,69,96)(64,95,70,89)(65,88,71,94)(66,93,72,87) );
G=PermutationGroup([[(1,83,7,77),(2,42,8,48),(3,73,9,79),(4,44,10,38),(5,75,11,81),(6,46,12,40),(13,31,19,25),(14,53,20,59),(15,33,21,27),(16,55,22,49),(17,35,23,29),(18,57,24,51),(26,65,32,71),(28,67,34,61),(30,69,36,63),(37,95,43,89),(39,85,45,91),(41,87,47,93),(50,68,56,62),(52,70,58,64),(54,72,60,66),(74,90,80,96),(76,92,82,86),(78,94,84,88)], [(1,70,87,13),(2,65,88,20),(3,72,89,15),(4,67,90,22),(5,62,91,17),(6,69,92,24),(7,64,93,19),(8,71,94,14),(9,66,95,21),(10,61,96,16),(11,68,85,23),(12,63,86,18),(25,77,52,41),(26,84,53,48),(27,79,54,43),(28,74,55,38),(29,81,56,45),(30,76,57,40),(31,83,58,47),(32,78,59,42),(33,73,60,37),(34,80,49,44),(35,75,50,39),(36,82,51,46)], [(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)], [(1,21,7,15),(2,14,8,20),(3,19,9,13),(4,24,10,18),(5,17,11,23),(6,22,12,16),(25,37,31,43),(26,42,32,48),(27,47,33,41),(28,40,34,46),(29,45,35,39),(30,38,36,44),(49,82,55,76),(50,75,56,81),(51,80,57,74),(52,73,58,79),(53,78,59,84),(54,83,60,77),(61,92,67,86),(62,85,68,91),(63,90,69,96),(64,95,70,89),(65,88,71,94),(66,93,72,87)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | 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 | D4○D12 |
| kernel | C42.163D6 | C42⋊2S3 | C4×D12 | Dic3⋊4D4 | Dic3⋊D4 | C23.21D6 | Dic3.Q8 | S3×C4⋊C4 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | D6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 4 | 2 |
Matrix representation of C42.163D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 0 | 8 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,8,0,0,0,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,1,0,0,0,0,0,8,0,0,0,0,0,0,5,12,0,0,0,0,11,8],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,8,0,0,0,0,3,12,0,0,0,0,0,0,5,0,0,0,0,0,11,8],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,2,5] >;
C42.163D6 in GAP, Magma, Sage, TeX
C_4^2._{163}D_6 % in TeX
G:=Group("C4^2.163D6"); // GroupNames label
G:=SmallGroup(192,1268);
// by ID
G=gap.SmallGroup(192,1268);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,184,1571,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations