metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.119D6, C6.1072+ 1+4, (C4×D4)⋊27S3, (C4×D12)⋊35C2, (D4×C12)⋊29C2, C4⋊C4.289D6, C12⋊7D4⋊13C2, (C2×D4).226D6, C23.9D6⋊11C2, C2.20(D4○D12), (C2×C6).109C24, C22⋊C4.121D6, C12.6Q8⋊17C2, (C22×C4).232D6, C23.14D6⋊28C2, Dic3⋊4D4⋊48C2, C12.293(C4○D4), (C4×C12).162C22, (C2×C12).588C23, D6⋊C4.144C22, (C6×D4).310C22, C22.2(C4○D12), C4.119(D4⋊2S3), (C2×D12).215C22, C23.26D6⋊10C2, Dic3⋊C4.67C22, (C22×S3).43C23, C4⋊Dic3.398C22, (C22×C12).84C22, C22.134(S3×C23), (C22×C6).179C23, C23.116(C22×S3), (C2×Dic3).49C23, C3⋊5(C22.47C24), (C4×Dic3).208C22, C6.D4.109C22, (C22×Dic3).101C22, C4⋊C4⋊S3⋊9C2, C6.51(C2×C4○D4), (C2×C4⋊Dic3)⋊26C2, C2.58(C2×C4○D12), (C2×C6).19(C4○D4), C2.25(C2×D4⋊2S3), (S3×C2×C4).205C22, (C3×C4⋊C4).337C22, (C2×C4).165(C22×S3), (C2×C3⋊D4).21C22, (C3×C22⋊C4).131C22, SmallGroup(192,1124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.119D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 600 in 238 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C22.47C24, C12.6Q8, C4×D12, Dic3⋊4D4, C23.9D6, C4⋊C4⋊S3, C2×C4⋊Dic3, C23.26D6, C12⋊7D4, C23.14D6, D4×C12, C42.119D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, D4⋊2S3, S3×C23, C22.47C24, C2×C4○D12, C2×D4⋊2S3, D4○D12, C42.119D6
►On 96 points
(1 65 37 67)(2 86 38 53)(3 61 39 69)(4 88 40 49)(5 63 41 71)(6 90 42 51)(7 91 59 81)(8 28 60 73)(9 93 55 83)(10 30 56 75)(11 95 57 79)(12 26 58 77)(13 27 43 78)(14 92 44 82)(15 29 45 74)(16 94 46 84)(17 25 47 76)(18 96 48 80)(19 54 35 87)(20 70 36 62)(21 50 31 89)(22 72 32 64)(23 52 33 85)(24 68 34 66)
(1 59 23 13)(2 60 24 14)(3 55 19 15)(4 56 20 16)(5 57 21 17)(6 58 22 18)(7 33 43 37)(8 34 44 38)(9 35 45 39)(10 36 46 40)(11 31 47 41)(12 32 48 42)(25 63 79 50)(26 64 80 51)(27 65 81 52)(28 66 82 53)(29 61 83 54)(30 62 84 49)(67 91 85 78)(68 92 86 73)(69 93 87 74)(70 94 88 75)(71 95 89 76)(72 96 90 77)
(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 37 42)(2 41 38 5)(3 4 39 40)(7 48 59 18)(8 17 60 47)(9 46 55 16)(10 15 56 45)(11 44 57 14)(12 13 58 43)(19 20 35 36)(21 24 31 34)(22 33 32 23)(25 82 76 92)(26 91 77 81)(27 80 78 96)(28 95 73 79)(29 84 74 94)(30 93 75 83)(49 87 88 54)(50 53 89 86)(51 85 90 52)(61 62 69 70)(63 66 71 68)(64 67 72 65)
G:=sub<Sym(96)| (1,65,37,67)(2,86,38,53)(3,61,39,69)(4,88,40,49)(5,63,41,71)(6,90,42,51)(7,91,59,81)(8,28,60,73)(9,93,55,83)(10,30,56,75)(11,95,57,79)(12,26,58,77)(13,27,43,78)(14,92,44,82)(15,29,45,74)(16,94,46,84)(17,25,47,76)(18,96,48,80)(19,54,35,87)(20,70,36,62)(21,50,31,89)(22,72,32,64)(23,52,33,85)(24,68,34,66), (1,59,23,13)(2,60,24,14)(3,55,19,15)(4,56,20,16)(5,57,21,17)(6,58,22,18)(7,33,43,37)(8,34,44,38)(9,35,45,39)(10,36,46,40)(11,31,47,41)(12,32,48,42)(25,63,79,50)(26,64,80,51)(27,65,81,52)(28,66,82,53)(29,61,83,54)(30,62,84,49)(67,91,85,78)(68,92,86,73)(69,93,87,74)(70,94,88,75)(71,95,89,76)(72,96,90,77), (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), (1,6,37,42)(2,41,38,5)(3,4,39,40)(7,48,59,18)(8,17,60,47)(9,46,55,16)(10,15,56,45)(11,44,57,14)(12,13,58,43)(19,20,35,36)(21,24,31,34)(22,33,32,23)(25,82,76,92)(26,91,77,81)(27,80,78,96)(28,95,73,79)(29,84,74,94)(30,93,75,83)(49,87,88,54)(50,53,89,86)(51,85,90,52)(61,62,69,70)(63,66,71,68)(64,67,72,65)>;
G:=Group( (1,65,37,67)(2,86,38,53)(3,61,39,69)(4,88,40,49)(5,63,41,71)(6,90,42,51)(7,91,59,81)(8,28,60,73)(9,93,55,83)(10,30,56,75)(11,95,57,79)(12,26,58,77)(13,27,43,78)(14,92,44,82)(15,29,45,74)(16,94,46,84)(17,25,47,76)(18,96,48,80)(19,54,35,87)(20,70,36,62)(21,50,31,89)(22,72,32,64)(23,52,33,85)(24,68,34,66), (1,59,23,13)(2,60,24,14)(3,55,19,15)(4,56,20,16)(5,57,21,17)(6,58,22,18)(7,33,43,37)(8,34,44,38)(9,35,45,39)(10,36,46,40)(11,31,47,41)(12,32,48,42)(25,63,79,50)(26,64,80,51)(27,65,81,52)(28,66,82,53)(29,61,83,54)(30,62,84,49)(67,91,85,78)(68,92,86,73)(69,93,87,74)(70,94,88,75)(71,95,89,76)(72,96,90,77), (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), (1,6,37,42)(2,41,38,5)(3,4,39,40)(7,48,59,18)(8,17,60,47)(9,46,55,16)(10,15,56,45)(11,44,57,14)(12,13,58,43)(19,20,35,36)(21,24,31,34)(22,33,32,23)(25,82,76,92)(26,91,77,81)(27,80,78,96)(28,95,73,79)(29,84,74,94)(30,93,75,83)(49,87,88,54)(50,53,89,86)(51,85,90,52)(61,62,69,70)(63,66,71,68)(64,67,72,65) );
G=PermutationGroup([[(1,65,37,67),(2,86,38,53),(3,61,39,69),(4,88,40,49),(5,63,41,71),(6,90,42,51),(7,91,59,81),(8,28,60,73),(9,93,55,83),(10,30,56,75),(11,95,57,79),(12,26,58,77),(13,27,43,78),(14,92,44,82),(15,29,45,74),(16,94,46,84),(17,25,47,76),(18,96,48,80),(19,54,35,87),(20,70,36,62),(21,50,31,89),(22,72,32,64),(23,52,33,85),(24,68,34,66)], [(1,59,23,13),(2,60,24,14),(3,55,19,15),(4,56,20,16),(5,57,21,17),(6,58,22,18),(7,33,43,37),(8,34,44,38),(9,35,45,39),(10,36,46,40),(11,31,47,41),(12,32,48,42),(25,63,79,50),(26,64,80,51),(27,65,81,52),(28,66,82,53),(29,61,83,54),(30,62,84,49),(67,91,85,78),(68,92,86,73),(69,93,87,74),(70,94,88,75),(71,95,89,76),(72,96,90,77)], [(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)], [(1,6,37,42),(2,41,38,5),(3,4,39,40),(7,48,59,18),(8,17,60,47),(9,46,55,16),(10,15,56,45),(11,44,57,14),(12,13,58,43),(19,20,35,36),(21,24,31,34),(22,33,32,23),(25,82,76,92),(26,91,77,81),(27,80,78,96),(28,95,73,79),(29,84,74,94),(30,93,75,83),(49,87,88,54),(50,53,89,86),(51,85,90,52),(61,62,69,70),(63,66,71,68),(64,67,72,65)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| 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 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ 1+4 | D4⋊2S3 | D4○D12 |
| kernel | C42.119D6 | C12.6Q8 | C4×D12 | Dic3⋊4D4 | C23.9D6 | C4⋊C4⋊S3 | C2×C4⋊Dic3 | C23.26D6 | C12⋊7D4 | C23.14D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C2×C6 | C22 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.119D6 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 10 | 6 |
| 0 | 0 | 7 | 3 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 11 | 11 |
| 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 4 | 11 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,10,7,0,0,6,3],[0,1,0,0,1,0,0,0,0,0,9,11,0,0,2,11],[0,1,0,0,12,0,0,0,0,0,2,4,0,0,2,11] >;
C42.119D6 in GAP, Magma, Sage, TeX
C_4^2._{119}D_6 % in TeX
G:=Group("C4^2.119D6"); // GroupNames label
G:=SmallGroup(192,1124);
// by ID
G=gap.SmallGroup(192,1124);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations