metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.154D6, C6.302- 1+4, C12⋊Q8⋊37C2, C4⋊C4.210D6, (C4×Dic6)⋊49C2, D6⋊Q8.3C2, C42.C2⋊10S3, Dic3.Q8⋊35C2, (C2×C6).240C24, C42⋊3S3.1C2, D6⋊C4.42C22, C2.59(Q8○D12), C4.Dic6⋊36C2, Dic6⋊C4⋊37C2, (C2×C12).602C23, (C4×C12).199C22, Dic3.13(C4○D4), C4⋊Dic3.316C22, C22.261(S3×C23), Dic3⋊C4.162C22, (C22×S3).105C23, C2.31(Q8.15D6), C3⋊4(C22.35C24), (C4×Dic3).216C22, (C2×Dic6).252C22, (C2×Dic3).260C23, C2.91(S3×C4○D4), C4⋊C4⋊S3.2C2, C6.202(C2×C4○D4), C4⋊C4⋊7S3.13C2, (S3×C2×C4).130C22, (C3×C42.C2)⋊13C2, (C3×C4⋊C4).195C22, (C2×C4).205(C22×S3), SmallGroup(192,1255)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.154D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c5 >
Subgroups: 416 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×C12, C22×S3, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22.35C24, C4×Dic6, C42⋊3S3, Dic6⋊C4, C12⋊Q8, Dic3.Q8, C4.Dic6, C4⋊C4⋊7S3, D6⋊Q8, C4⋊C4⋊S3, C3×C42.C2, C42.154D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, S3×C23, C22.35C24, Q8.15D6, S3×C4○D4, Q8○D12, C42.154D6
►On 96 points
Generators in S96
(1 58 7 52)(2 76 8 82)(3 60 9 54)(4 78 10 84)(5 50 11 56)(6 80 12 74)(13 79 19 73)(14 51 20 57)(15 81 21 75)(16 53 22 59)(17 83 23 77)(18 55 24 49)(25 47 31 41)(26 67 32 61)(27 37 33 43)(28 69 34 63)(29 39 35 45)(30 71 36 65)(38 93 44 87)(40 95 46 89)(42 85 48 91)(62 86 68 92)(64 88 70 94)(66 90 72 96)
(1 85 15 32)(2 33 16 86)(3 87 17 34)(4 35 18 88)(5 89 19 36)(6 25 20 90)(7 91 21 26)(8 27 22 92)(9 93 23 28)(10 29 24 94)(11 95 13 30)(12 31 14 96)(37 59 62 82)(38 83 63 60)(39 49 64 84)(40 73 65 50)(41 51 66 74)(42 75 67 52)(43 53 68 76)(44 77 69 54)(45 55 70 78)(46 79 71 56)(47 57 72 80)(48 81 61 58)
(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 22 19 16)(14 15 20 21)(17 18 23 24)(25 85 31 91)(26 90 32 96)(27 95 33 89)(28 88 34 94)(29 93 35 87)(30 86 36 92)(37 40 43 46)(38 45 44 39)(41 48 47 42)(49 77 55 83)(50 82 56 76)(51 75 57 81)(52 80 58 74)(53 73 59 79)(54 78 60 84)(61 72 67 66)(62 65 68 71)(63 70 69 64)
G:=sub<Sym(96)| (1,58,7,52)(2,76,8,82)(3,60,9,54)(4,78,10,84)(5,50,11,56)(6,80,12,74)(13,79,19,73)(14,51,20,57)(15,81,21,75)(16,53,22,59)(17,83,23,77)(18,55,24,49)(25,47,31,41)(26,67,32,61)(27,37,33,43)(28,69,34,63)(29,39,35,45)(30,71,36,65)(38,93,44,87)(40,95,46,89)(42,85,48,91)(62,86,68,92)(64,88,70,94)(66,90,72,96), (1,85,15,32)(2,33,16,86)(3,87,17,34)(4,35,18,88)(5,89,19,36)(6,25,20,90)(7,91,21,26)(8,27,22,92)(9,93,23,28)(10,29,24,94)(11,95,13,30)(12,31,14,96)(37,59,62,82)(38,83,63,60)(39,49,64,84)(40,73,65,50)(41,51,66,74)(42,75,67,52)(43,53,68,76)(44,77,69,54)(45,55,70,78)(46,79,71,56)(47,57,72,80)(48,81,61,58), (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)(13,22,19,16)(14,15,20,21)(17,18,23,24)(25,85,31,91)(26,90,32,96)(27,95,33,89)(28,88,34,94)(29,93,35,87)(30,86,36,92)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,77,55,83)(50,82,56,76)(51,75,57,81)(52,80,58,74)(53,73,59,79)(54,78,60,84)(61,72,67,66)(62,65,68,71)(63,70,69,64)>;
G:=Group( (1,58,7,52)(2,76,8,82)(3,60,9,54)(4,78,10,84)(5,50,11,56)(6,80,12,74)(13,79,19,73)(14,51,20,57)(15,81,21,75)(16,53,22,59)(17,83,23,77)(18,55,24,49)(25,47,31,41)(26,67,32,61)(27,37,33,43)(28,69,34,63)(29,39,35,45)(30,71,36,65)(38,93,44,87)(40,95,46,89)(42,85,48,91)(62,86,68,92)(64,88,70,94)(66,90,72,96), (1,85,15,32)(2,33,16,86)(3,87,17,34)(4,35,18,88)(5,89,19,36)(6,25,20,90)(7,91,21,26)(8,27,22,92)(9,93,23,28)(10,29,24,94)(11,95,13,30)(12,31,14,96)(37,59,62,82)(38,83,63,60)(39,49,64,84)(40,73,65,50)(41,51,66,74)(42,75,67,52)(43,53,68,76)(44,77,69,54)(45,55,70,78)(46,79,71,56)(47,57,72,80)(48,81,61,58), (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)(13,22,19,16)(14,15,20,21)(17,18,23,24)(25,85,31,91)(26,90,32,96)(27,95,33,89)(28,88,34,94)(29,93,35,87)(30,86,36,92)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,77,55,83)(50,82,56,76)(51,75,57,81)(52,80,58,74)(53,73,59,79)(54,78,60,84)(61,72,67,66)(62,65,68,71)(63,70,69,64) );
G=PermutationGroup([[(1,58,7,52),(2,76,8,82),(3,60,9,54),(4,78,10,84),(5,50,11,56),(6,80,12,74),(13,79,19,73),(14,51,20,57),(15,81,21,75),(16,53,22,59),(17,83,23,77),(18,55,24,49),(25,47,31,41),(26,67,32,61),(27,37,33,43),(28,69,34,63),(29,39,35,45),(30,71,36,65),(38,93,44,87),(40,95,46,89),(42,85,48,91),(62,86,68,92),(64,88,70,94),(66,90,72,96)], [(1,85,15,32),(2,33,16,86),(3,87,17,34),(4,35,18,88),(5,89,19,36),(6,25,20,90),(7,91,21,26),(8,27,22,92),(9,93,23,28),(10,29,24,94),(11,95,13,30),(12,31,14,96),(37,59,62,82),(38,83,63,60),(39,49,64,84),(40,73,65,50),(41,51,66,74),(42,75,67,52),(43,53,68,76),(44,77,69,54),(45,55,70,78),(46,79,71,56),(47,57,72,80),(48,81,61,58)], [(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),(13,22,19,16),(14,15,20,21),(17,18,23,24),(25,85,31,91),(26,90,32,96),(27,95,33,89),(28,88,34,94),(29,93,35,87),(30,86,36,92),(37,40,43,46),(38,45,44,39),(41,48,47,42),(49,77,55,83),(50,82,56,76),(51,75,57,81),(52,80,58,74),(53,73,59,79),(54,78,60,84),(61,72,67,66),(62,65,68,71),(63,70,69,64)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2- 1+4 | Q8.15D6 | S3×C4○D4 | Q8○D12 |
| kernel | C42.154D6 | C4×Dic6 | C42⋊3S3 | Dic6⋊C4 | C12⋊Q8 | Dic3.Q8 | C4.Dic6 | C4⋊C4⋊7S3 | D6⋊Q8 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | Dic3 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 6 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of C42.154D6 ►in GL6(𝔽13)
| 5 | 3 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [5,5,0,0,0,0,3,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,2,1,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,5,8,0,0,0,0,5,0,0,0,0,0,0,0,8,5,0,0,0,0,8,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,5,8,0,0,0,0,0,0,8,0,0,0,0,0,8,5] >;
C42.154D6 in GAP, Magma, Sage, TeX
C_4^2._{154}D_6 % in TeX
G:=Group("C4^2.154D6"); // GroupNames label
G:=SmallGroup(192,1255);
// by ID
G=gap.SmallGroup(192,1255);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,555,100,1571,570,136,6278]);
// 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=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations