metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.148D6, C6.952- 1+4, C6.1302+ 1+4, (C4×S3)⋊1Q8, C12⋊Q8⋊35C2, D6.4(C2×Q8), C4.39(S3×Q8), C4⋊C4.111D6, C12.50(C2×Q8), C42.C2⋊4S3, C12⋊2Q8⋊32C2, D6⋊Q8.2C2, Dic3.6(C2×Q8), C2.55(D4○D12), C6.42(C22×Q8), Dic3.Q8⋊32C2, (C2×C6).233C24, (C2×C12).87C23, C42⋊2S3.6C2, D6⋊C4.39C22, C4.D12.11C2, C2.57(Q8○D12), C4.Dic6⋊33C2, (C4×C12).193C22, C4⋊Dic3.240C22, C22.254(S3×C23), (C2×Dic6).40C22, Dic3⋊C4.122C22, (C22×S3).220C23, C3⋊4(C23.41C23), (C2×Dic3).121C23, (C4×Dic3).140C22, C2.25(C2×S3×Q8), (S3×C4⋊C4).11C2, C4⋊C4⋊7S3.12C2, (C3×C42.C2)⋊6C2, (S3×C2×C4).124C22, (C3×C4⋊C4).188C22, (C2×C4).203(C22×S3), SmallGroup(192,1248)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.148D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >
Subgroups: 480 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C23.41C23, C12⋊2Q8, C42⋊2S3, C12⋊Q8, C12⋊Q8, Dic3.Q8, C4.Dic6, S3×C4⋊C4, C4⋊C4⋊7S3, D6⋊Q8, C4.D12, C3×C42.C2, C42.148D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C22×S3, C22×Q8, 2+ 1+4, 2- 1+4, S3×Q8, S3×C23, C23.41C23, C2×S3×Q8, D4○D12, Q8○D12, C42.148D6
►On 96 points
(1 20 70 48)(2 37 71 21)(3 22 72 38)(4 39 61 23)(5 24 62 40)(6 41 63 13)(7 14 64 42)(8 43 65 15)(9 16 66 44)(10 45 67 17)(11 18 68 46)(12 47 69 19)(25 53 87 77)(26 78 88 54)(27 55 89 79)(28 80 90 56)(29 57 91 81)(30 82 92 58)(31 59 93 83)(32 84 94 60)(33 49 95 73)(34 74 96 50)(35 51 85 75)(36 76 86 52)
(1 35 7 29)(2 92 8 86)(3 25 9 31)(4 94 10 88)(5 27 11 33)(6 96 12 90)(13 74 19 80)(14 57 20 51)(15 76 21 82)(16 59 22 53)(17 78 23 84)(18 49 24 55)(26 61 32 67)(28 63 34 69)(30 65 36 71)(37 58 43 52)(38 77 44 83)(39 60 45 54)(40 79 46 73)(41 50 47 56)(42 81 48 75)(62 89 68 95)(64 91 70 85)(66 93 72 87)
(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 88 31 94)(26 93 32 87)(27 86 33 92)(28 91 34 85)(29 96 35 90)(30 89 36 95)(37 46 43 40)(38 39 44 45)(41 42 47 48)(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 66 67 72)(62 71 68 65)(63 64 69 70)
G:=sub<Sym(96)| (1,20,70,48)(2,37,71,21)(3,22,72,38)(4,39,61,23)(5,24,62,40)(6,41,63,13)(7,14,64,42)(8,43,65,15)(9,16,66,44)(10,45,67,17)(11,18,68,46)(12,47,69,19)(25,53,87,77)(26,78,88,54)(27,55,89,79)(28,80,90,56)(29,57,91,81)(30,82,92,58)(31,59,93,83)(32,84,94,60)(33,49,95,73)(34,74,96,50)(35,51,85,75)(36,76,86,52), (1,35,7,29)(2,92,8,86)(3,25,9,31)(4,94,10,88)(5,27,11,33)(6,96,12,90)(13,74,19,80)(14,57,20,51)(15,76,21,82)(16,59,22,53)(17,78,23,84)(18,49,24,55)(26,61,32,67)(28,63,34,69)(30,65,36,71)(37,58,43,52)(38,77,44,83)(39,60,45,54)(40,79,46,73)(41,50,47,56)(42,81,48,75)(62,89,68,95)(64,91,70,85)(66,93,72,87), (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)(25,88,31,94)(26,93,32,87)(27,86,33,92)(28,91,34,85)(29,96,35,90)(30,89,36,95)(37,46,43,40)(38,39,44,45)(41,42,47,48)(49,82,55,76)(50,75,56,81)(51,80,57,74)(52,73,58,79)(53,78,59,84)(54,83,60,77)(61,66,67,72)(62,71,68,65)(63,64,69,70)>;
G:=Group( (1,20,70,48)(2,37,71,21)(3,22,72,38)(4,39,61,23)(5,24,62,40)(6,41,63,13)(7,14,64,42)(8,43,65,15)(9,16,66,44)(10,45,67,17)(11,18,68,46)(12,47,69,19)(25,53,87,77)(26,78,88,54)(27,55,89,79)(28,80,90,56)(29,57,91,81)(30,82,92,58)(31,59,93,83)(32,84,94,60)(33,49,95,73)(34,74,96,50)(35,51,85,75)(36,76,86,52), (1,35,7,29)(2,92,8,86)(3,25,9,31)(4,94,10,88)(5,27,11,33)(6,96,12,90)(13,74,19,80)(14,57,20,51)(15,76,21,82)(16,59,22,53)(17,78,23,84)(18,49,24,55)(26,61,32,67)(28,63,34,69)(30,65,36,71)(37,58,43,52)(38,77,44,83)(39,60,45,54)(40,79,46,73)(41,50,47,56)(42,81,48,75)(62,89,68,95)(64,91,70,85)(66,93,72,87), (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)(25,88,31,94)(26,93,32,87)(27,86,33,92)(28,91,34,85)(29,96,35,90)(30,89,36,95)(37,46,43,40)(38,39,44,45)(41,42,47,48)(49,82,55,76)(50,75,56,81)(51,80,57,74)(52,73,58,79)(53,78,59,84)(54,83,60,77)(61,66,67,72)(62,71,68,65)(63,64,69,70) );
G=PermutationGroup([[(1,20,70,48),(2,37,71,21),(3,22,72,38),(4,39,61,23),(5,24,62,40),(6,41,63,13),(7,14,64,42),(8,43,65,15),(9,16,66,44),(10,45,67,17),(11,18,68,46),(12,47,69,19),(25,53,87,77),(26,78,88,54),(27,55,89,79),(28,80,90,56),(29,57,91,81),(30,82,92,58),(31,59,93,83),(32,84,94,60),(33,49,95,73),(34,74,96,50),(35,51,85,75),(36,76,86,52)], [(1,35,7,29),(2,92,8,86),(3,25,9,31),(4,94,10,88),(5,27,11,33),(6,96,12,90),(13,74,19,80),(14,57,20,51),(15,76,21,82),(16,59,22,53),(17,78,23,84),(18,49,24,55),(26,61,32,67),(28,63,34,69),(30,65,36,71),(37,58,43,52),(38,77,44,83),(39,60,45,54),(40,79,46,73),(41,50,47,56),(42,81,48,75),(62,89,68,95),(64,91,70,85),(66,93,72,87)], [(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),(25,88,31,94),(26,93,32,87),(27,86,33,92),(28,91,34,85),(29,96,35,90),(30,89,36,95),(37,46,43,40),(38,39,44,45),(41,42,47,48),(49,82,55,76),(50,75,56,81),(51,80,57,74),(52,73,58,79),(53,78,59,84),(54,83,60,77),(61,66,67,72),(62,71,68,65),(63,64,69,70)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | 2+ 1+4 | 2- 1+4 | S3×Q8 | D4○D12 | Q8○D12 |
| kernel | C42.148D6 | C12⋊2Q8 | C42⋊2S3 | C12⋊Q8 | Dic3.Q8 | C4.Dic6 | S3×C4⋊C4 | C4⋊C4⋊7S3 | D6⋊Q8 | C4.D12 | C3×C42.C2 | C42.C2 | C4×S3 | C42 | C4⋊C4 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 1 | 6 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C42.148D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,8,0,0,0,0,0,0,8,0,0],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,12,1,0,0,0,0,12,0,0,0],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,1,0,0,0,0,0,1,12,0,0] >;
C42.148D6 in GAP, Magma, Sage, TeX
C_4^2._{148}D_6 % in TeX
G:=Group("C4^2.148D6"); // GroupNames label
G:=SmallGroup(192,1248);
// by ID
G=gap.SmallGroup(192,1248);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,297,192,6278]);
// 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^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations