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, Dic3⋊4D4⋊48C2, C23.14D6⋊28C2, C12.293(C4○D4), (C2×C12).588C23, (C4×C12).162C22, 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×C6).179C23, C23.116(C22×S3), C22.134(S3×C23), (C22×C12).84C22, (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
Subgroups: 600 in 238 conjugacy classes, 99 normal (51 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], S3 [×2], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], Dic3 [×6], C12 [×2], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], C4×S3 [×2], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×6], C2×C12 [×5], C2×C12 [×4], C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×3], C4⋊Dic3 [×2], D6⋊C4 [×6], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C22×Dic3 [×2], C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, C22.47C24, C12.6Q8, C4×D12, Dic3⋊4D4 [×2], C23.9D6 [×2], C4⋊C4⋊S3 [×2], C2×C4⋊Dic3, C23.26D6, C12⋊7D4 [×2], C23.14D6 [×2], D4×C12, C42.119D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), C4○D12 [×2], D4⋊2S3 [×2], S3×C23, C22.47C24, C2×C4○D12, C2×D4⋊2S3, D4○D12, C42.119D6
Generators and relations
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 >
►On 96 points
(1 65 37 71)(2 86 38 51)(3 61 39 67)(4 88 40 53)(5 63 41 69)(6 90 42 49)(7 95 57 83)(8 26 58 75)(9 91 59 79)(10 28 60 77)(11 93 55 81)(12 30 56 73)(13 29 45 78)(14 94 46 82)(15 25 47 74)(16 96 48 84)(17 27 43 76)(18 92 44 80)(19 52 33 87)(20 68 34 62)(21 54 35 89)(22 70 36 64)(23 50 31 85)(24 72 32 66)
(1 59 23 17)(2 60 24 18)(3 55 19 13)(4 56 20 14)(5 57 21 15)(6 58 22 16)(7 35 47 41)(8 36 48 42)(9 31 43 37)(10 32 44 38)(11 33 45 39)(12 34 46 40)(25 63 83 54)(26 64 84 49)(27 65 79 50)(28 66 80 51)(29 61 81 52)(30 62 82 53)(67 93 87 78)(68 94 88 73)(69 95 89 74)(70 96 90 75)(71 91 85 76)(72 92 86 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 44 57 18)(8 17 58 43)(9 48 59 16)(10 15 60 47)(11 46 55 14)(12 13 56 45)(19 20 33 34)(21 24 35 32)(22 31 36 23)(25 80 74 92)(26 91 75 79)(27 84 76 96)(28 95 77 83)(29 82 78 94)(30 93 73 81)(49 85 90 50)(51 89 86 54)(52 53 87 88)(61 62 67 68)(63 66 69 72)(64 71 70 65)
G:=sub<Sym(96)| (1,65,37,71)(2,86,38,51)(3,61,39,67)(4,88,40,53)(5,63,41,69)(6,90,42,49)(7,95,57,83)(8,26,58,75)(9,91,59,79)(10,28,60,77)(11,93,55,81)(12,30,56,73)(13,29,45,78)(14,94,46,82)(15,25,47,74)(16,96,48,84)(17,27,43,76)(18,92,44,80)(19,52,33,87)(20,68,34,62)(21,54,35,89)(22,70,36,64)(23,50,31,85)(24,72,32,66), (1,59,23,17)(2,60,24,18)(3,55,19,13)(4,56,20,14)(5,57,21,15)(6,58,22,16)(7,35,47,41)(8,36,48,42)(9,31,43,37)(10,32,44,38)(11,33,45,39)(12,34,46,40)(25,63,83,54)(26,64,84,49)(27,65,79,50)(28,66,80,51)(29,61,81,52)(30,62,82,53)(67,93,87,78)(68,94,88,73)(69,95,89,74)(70,96,90,75)(71,91,85,76)(72,92,86,77), (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)(7,44,57,18)(8,17,58,43)(9,48,59,16)(10,15,60,47)(11,46,55,14)(12,13,56,45)(19,20,33,34)(21,24,35,32)(22,31,36,23)(25,80,74,92)(26,91,75,79)(27,84,76,96)(28,95,77,83)(29,82,78,94)(30,93,73,81)(49,85,90,50)(51,89,86,54)(52,53,87,88)(61,62,67,68)(63,66,69,72)(64,71,70,65)>;
G:=Group( (1,65,37,71)(2,86,38,51)(3,61,39,67)(4,88,40,53)(5,63,41,69)(6,90,42,49)(7,95,57,83)(8,26,58,75)(9,91,59,79)(10,28,60,77)(11,93,55,81)(12,30,56,73)(13,29,45,78)(14,94,46,82)(15,25,47,74)(16,96,48,84)(17,27,43,76)(18,92,44,80)(19,52,33,87)(20,68,34,62)(21,54,35,89)(22,70,36,64)(23,50,31,85)(24,72,32,66), (1,59,23,17)(2,60,24,18)(3,55,19,13)(4,56,20,14)(5,57,21,15)(6,58,22,16)(7,35,47,41)(8,36,48,42)(9,31,43,37)(10,32,44,38)(11,33,45,39)(12,34,46,40)(25,63,83,54)(26,64,84,49)(27,65,79,50)(28,66,80,51)(29,61,81,52)(30,62,82,53)(67,93,87,78)(68,94,88,73)(69,95,89,74)(70,96,90,75)(71,91,85,76)(72,92,86,77), (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)(7,44,57,18)(8,17,58,43)(9,48,59,16)(10,15,60,47)(11,46,55,14)(12,13,56,45)(19,20,33,34)(21,24,35,32)(22,31,36,23)(25,80,74,92)(26,91,75,79)(27,84,76,96)(28,95,77,83)(29,82,78,94)(30,93,73,81)(49,85,90,50)(51,89,86,54)(52,53,87,88)(61,62,67,68)(63,66,69,72)(64,71,70,65) );
G=PermutationGroup([(1,65,37,71),(2,86,38,51),(3,61,39,67),(4,88,40,53),(5,63,41,69),(6,90,42,49),(7,95,57,83),(8,26,58,75),(9,91,59,79),(10,28,60,77),(11,93,55,81),(12,30,56,73),(13,29,45,78),(14,94,46,82),(15,25,47,74),(16,96,48,84),(17,27,43,76),(18,92,44,80),(19,52,33,87),(20,68,34,62),(21,54,35,89),(22,70,36,64),(23,50,31,85),(24,72,32,66)], [(1,59,23,17),(2,60,24,18),(3,55,19,13),(4,56,20,14),(5,57,21,15),(6,58,22,16),(7,35,47,41),(8,36,48,42),(9,31,43,37),(10,32,44,38),(11,33,45,39),(12,34,46,40),(25,63,83,54),(26,64,84,49),(27,65,79,50),(28,66,80,51),(29,61,81,52),(30,62,82,53),(67,93,87,78),(68,94,88,73),(69,95,89,74),(70,96,90,75),(71,91,85,76),(72,92,86,77)], [(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),(7,44,57,18),(8,17,58,43),(9,48,59,16),(10,15,60,47),(11,46,55,14),(12,13,56,45),(19,20,33,34),(21,24,35,32),(22,31,36,23),(25,80,74,92),(26,91,75,79),(27,84,76,96),(28,95,77,83),(29,82,78,94),(30,93,73,81),(49,85,90,50),(51,89,86,54),(52,53,87,88),(61,62,67,68),(63,66,69,72),(64,71,70,65)])
Matrix representation ►G ⊆ 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] >;
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 |
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