metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.133D6, C6.132- 1+4, C6.1122+ 1+4, C12⋊Q8⋊17C2, (C4×Q8)⋊19S3, (C4×D12)⋊41C2, C4⋊C4.300D6, (Q8×C12)⋊17C2, D6⋊3Q8⋊10C2, D6⋊Q8⋊12C2, D6.D4⋊9C2, D6⋊C4.7C22, (C2×Q8).205D6, C42⋊2S3⋊18C2, C42⋊3S3⋊12C2, C42⋊7S3⋊29C2, C12⋊D4.10C2, C2.24(D4○D12), C4.49(C4○D12), C12.23D4⋊9C2, (C2×C6).126C24, C12.120(C4○D4), (C4×C12).178C22, (C2×C12).171C23, (C6×Q8).226C22, (C2×D12).263C22, Dic3⋊C4.77C22, (C22×S3).48C23, C4⋊Dic3.369C22, C22.147(S3×C23), (C2×Dic3).57C23, (C2×Dic6).32C22, (C4×Dic3).86C22, C2.14(Q8.15D6), C3⋊3(C22.36C24), C6.56(C2×C4○D4), C2.65(C2×C4○D12), (S3×C2×C4).76C22, (C3×C4⋊C4).354C22, (C2×C4).171(C22×S3), SmallGroup(192,1141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.133D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c5 >
Subgroups: 568 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C6×Q8, C22.36C24, C42⋊2S3, C4×D12, C42⋊7S3, C42⋊3S3, C12⋊Q8, D6.D4, C12⋊D4, D6⋊Q8, D6⋊3Q8, C12.23D4, Q8×C12, C42.133D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, C4○D12, S3×C23, C22.36C24, C2×C4○D12, Q8.15D6, D4○D12, C42.133D6
►On 96 points
(1 80 7 74)(2 75 8 81)(3 82 9 76)(4 77 10 83)(5 84 11 78)(6 79 12 73)(13 64 19 70)(14 71 20 65)(15 66 21 72)(16 61 22 67)(17 68 23 62)(18 63 24 69)(25 90 31 96)(26 85 32 91)(27 92 33 86)(28 87 34 93)(29 94 35 88)(30 89 36 95)(37 56 43 50)(38 51 44 57)(39 58 45 52)(40 53 46 59)(41 60 47 54)(42 55 48 49)
(1 52 36 67)(2 53 25 68)(3 54 26 69)(4 55 27 70)(5 56 28 71)(6 57 29 72)(7 58 30 61)(8 59 31 62)(9 60 32 63)(10 49 33 64)(11 50 34 65)(12 51 35 66)(13 77 48 92)(14 78 37 93)(15 79 38 94)(16 80 39 95)(17 81 40 96)(18 82 41 85)(19 83 42 86)(20 84 43 87)(21 73 44 88)(22 74 45 89)(23 75 46 90)(24 76 47 91)
(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 24 30 41)(2 40 31 23)(3 22 32 39)(4 38 33 21)(5 20 34 37)(6 48 35 19)(7 18 36 47)(8 46 25 17)(9 16 26 45)(10 44 27 15)(11 14 28 43)(12 42 29 13)(49 94 70 73)(50 84 71 93)(51 92 72 83)(52 82 61 91)(53 90 62 81)(54 80 63 89)(55 88 64 79)(56 78 65 87)(57 86 66 77)(58 76 67 85)(59 96 68 75)(60 74 69 95)
G:=sub<Sym(96)| (1,80,7,74)(2,75,8,81)(3,82,9,76)(4,77,10,83)(5,84,11,78)(6,79,12,73)(13,64,19,70)(14,71,20,65)(15,66,21,72)(16,61,22,67)(17,68,23,62)(18,63,24,69)(25,90,31,96)(26,85,32,91)(27,92,33,86)(28,87,34,93)(29,94,35,88)(30,89,36,95)(37,56,43,50)(38,51,44,57)(39,58,45,52)(40,53,46,59)(41,60,47,54)(42,55,48,49), (1,52,36,67)(2,53,25,68)(3,54,26,69)(4,55,27,70)(5,56,28,71)(6,57,29,72)(7,58,30,61)(8,59,31,62)(9,60,32,63)(10,49,33,64)(11,50,34,65)(12,51,35,66)(13,77,48,92)(14,78,37,93)(15,79,38,94)(16,80,39,95)(17,81,40,96)(18,82,41,85)(19,83,42,86)(20,84,43,87)(21,73,44,88)(22,74,45,89)(23,75,46,90)(24,76,47,91), (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), (1,24,30,41)(2,40,31,23)(3,22,32,39)(4,38,33,21)(5,20,34,37)(6,48,35,19)(7,18,36,47)(8,46,25,17)(9,16,26,45)(10,44,27,15)(11,14,28,43)(12,42,29,13)(49,94,70,73)(50,84,71,93)(51,92,72,83)(52,82,61,91)(53,90,62,81)(54,80,63,89)(55,88,64,79)(56,78,65,87)(57,86,66,77)(58,76,67,85)(59,96,68,75)(60,74,69,95)>;
G:=Group( (1,80,7,74)(2,75,8,81)(3,82,9,76)(4,77,10,83)(5,84,11,78)(6,79,12,73)(13,64,19,70)(14,71,20,65)(15,66,21,72)(16,61,22,67)(17,68,23,62)(18,63,24,69)(25,90,31,96)(26,85,32,91)(27,92,33,86)(28,87,34,93)(29,94,35,88)(30,89,36,95)(37,56,43,50)(38,51,44,57)(39,58,45,52)(40,53,46,59)(41,60,47,54)(42,55,48,49), (1,52,36,67)(2,53,25,68)(3,54,26,69)(4,55,27,70)(5,56,28,71)(6,57,29,72)(7,58,30,61)(8,59,31,62)(9,60,32,63)(10,49,33,64)(11,50,34,65)(12,51,35,66)(13,77,48,92)(14,78,37,93)(15,79,38,94)(16,80,39,95)(17,81,40,96)(18,82,41,85)(19,83,42,86)(20,84,43,87)(21,73,44,88)(22,74,45,89)(23,75,46,90)(24,76,47,91), (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), (1,24,30,41)(2,40,31,23)(3,22,32,39)(4,38,33,21)(5,20,34,37)(6,48,35,19)(7,18,36,47)(8,46,25,17)(9,16,26,45)(10,44,27,15)(11,14,28,43)(12,42,29,13)(49,94,70,73)(50,84,71,93)(51,92,72,83)(52,82,61,91)(53,90,62,81)(54,80,63,89)(55,88,64,79)(56,78,65,87)(57,86,66,77)(58,76,67,85)(59,96,68,75)(60,74,69,95) );
G=PermutationGroup([[(1,80,7,74),(2,75,8,81),(3,82,9,76),(4,77,10,83),(5,84,11,78),(6,79,12,73),(13,64,19,70),(14,71,20,65),(15,66,21,72),(16,61,22,67),(17,68,23,62),(18,63,24,69),(25,90,31,96),(26,85,32,91),(27,92,33,86),(28,87,34,93),(29,94,35,88),(30,89,36,95),(37,56,43,50),(38,51,44,57),(39,58,45,52),(40,53,46,59),(41,60,47,54),(42,55,48,49)], [(1,52,36,67),(2,53,25,68),(3,54,26,69),(4,55,27,70),(5,56,28,71),(6,57,29,72),(7,58,30,61),(8,59,31,62),(9,60,32,63),(10,49,33,64),(11,50,34,65),(12,51,35,66),(13,77,48,92),(14,78,37,93),(15,79,38,94),(16,80,39,95),(17,81,40,96),(18,82,41,85),(19,83,42,86),(20,84,43,87),(21,73,44,88),(22,74,45,89),(23,75,46,90),(24,76,47,91)], [(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)], [(1,24,30,41),(2,40,31,23),(3,22,32,39),(4,38,33,21),(5,20,34,37),(6,48,35,19),(7,18,36,47),(8,46,25,17),(9,16,26,45),(10,44,27,15),(11,14,28,43),(12,42,29,13),(49,94,70,73),(50,84,71,93),(51,92,72,83),(52,82,61,91),(53,90,62,81),(54,80,63,89),(55,88,64,79),(56,78,65,87),(57,86,66,77),(58,76,67,85),(59,96,68,75),(60,74,69,95)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 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 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | 2- 1+4 | Q8.15D6 | D4○D12 |
| kernel | C42.133D6 | C42⋊2S3 | C4×D12 | C42⋊7S3 | C42⋊3S3 | C12⋊Q8 | D6.D4 | C12⋊D4 | D6⋊Q8 | D6⋊3Q8 | C12.23D4 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C42.133D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 10 | 12 | 11 |
| 0 | 0 | 3 | 11 | 2 | 1 |
| 0 | 0 | 4 | 0 | 5 | 3 |
| 0 | 0 | 0 | 4 | 10 | 2 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 2 | 2 | 2 |
| 0 | 0 | 11 | 9 | 11 | 0 |
| 0 | 0 | 12 | 12 | 2 | 11 |
| 0 | 0 | 1 | 0 | 2 | 4 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 10 | 4 | 4 |
| 0 | 0 | 8 | 11 | 0 | 9 |
| 0 | 0 | 2 | 2 | 3 | 11 |
| 0 | 0 | 0 | 11 | 8 | 10 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,3,4,0,0,0,10,11,0,4,0,0,12,2,5,10,0,0,11,1,3,2],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,11,12,1,0,0,2,9,12,0,0,0,2,11,2,2,0,0,2,0,11,4],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,2,8,2,0,0,0,10,11,2,11,0,0,4,0,3,8,0,0,4,9,11,10] >;
C42.133D6 in GAP, Magma, Sage, TeX
C_4^2._{133}D_6 % in TeX
G:=Group("C4^2.133D6"); // GroupNames label
G:=SmallGroup(192,1141);
// by ID
G=gap.SmallGroup(192,1141);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,100,675,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^5>;
// generators/relations