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, C4.49(C4○D12), C2.24(D4○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×Dic6).32C22, (C2×Dic3).57C23, (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
Subgroups: 568 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×11], C22, C22 [×9], S3 [×3], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×9], D4 [×4], Q8 [×4], C23 [×3], Dic3 [×5], C12 [×2], C12 [×6], D6 [×9], C2×C6, C42, C42 [×2], C42, C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2×Q8 [×2], Dic6 [×2], C4×S3 [×4], D12 [×4], C2×Dic3 [×3], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3, C22×S3 [×2], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, C4×Dic3, Dic3⋊C4 [×2], Dic3⋊C4 [×4], C4⋊Dic3, D6⋊C4 [×2], D6⋊C4 [×10], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], C2×D12, C2×D12 [×2], C6×Q8, C22.36C24, C42⋊2S3, C4×D12, C42⋊7S3 [×2], C42⋊3S3 [×2], C12⋊Q8, D6.D4 [×2], C12⋊D4, D6⋊Q8 [×2], D6⋊3Q8, C12.23D4, Q8×C12, C42.133D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C4○D12 [×2], S3×C23, C22.36C24, C2×C4○D12, Q8.15D6, D4○D12, C42.133D6
Generators and relations
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 >
►On 96 points
(1 78 7 84)(2 73 8 79)(3 80 9 74)(4 75 10 81)(5 82 11 76)(6 77 12 83)(13 38 19 44)(14 45 20 39)(15 40 21 46)(16 47 22 41)(17 42 23 48)(18 37 24 43)(25 67 31 61)(26 62 32 68)(27 69 33 63)(28 64 34 70)(29 71 35 65)(30 66 36 72)(49 94 55 88)(50 89 56 95)(51 96 57 90)(52 91 58 85)(53 86 59 92)(54 93 60 87)
(1 43 67 55)(2 44 68 56)(3 45 69 57)(4 46 70 58)(5 47 71 59)(6 48 72 60)(7 37 61 49)(8 38 62 50)(9 39 63 51)(10 40 64 52)(11 41 65 53)(12 42 66 54)(13 26 95 73)(14 27 96 74)(15 28 85 75)(16 29 86 76)(17 30 87 77)(18 31 88 78)(19 32 89 79)(20 33 90 80)(21 34 91 81)(22 35 92 82)(23 36 93 83)(24 25 94 84)
(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 96 61 20)(2 19 62 95)(3 94 63 18)(4 17 64 93)(5 92 65 16)(6 15 66 91)(7 90 67 14)(8 13 68 89)(9 88 69 24)(10 23 70 87)(11 86 71 22)(12 21 72 85)(25 45 78 51)(26 50 79 44)(27 43 80 49)(28 60 81 42)(29 41 82 59)(30 58 83 40)(31 39 84 57)(32 56 73 38)(33 37 74 55)(34 54 75 48)(35 47 76 53)(36 52 77 46)
G:=sub<Sym(96)| (1,78,7,84)(2,73,8,79)(3,80,9,74)(4,75,10,81)(5,82,11,76)(6,77,12,83)(13,38,19,44)(14,45,20,39)(15,40,21,46)(16,47,22,41)(17,42,23,48)(18,37,24,43)(25,67,31,61)(26,62,32,68)(27,69,33,63)(28,64,34,70)(29,71,35,65)(30,66,36,72)(49,94,55,88)(50,89,56,95)(51,96,57,90)(52,91,58,85)(53,86,59,92)(54,93,60,87), (1,43,67,55)(2,44,68,56)(3,45,69,57)(4,46,70,58)(5,47,71,59)(6,48,72,60)(7,37,61,49)(8,38,62,50)(9,39,63,51)(10,40,64,52)(11,41,65,53)(12,42,66,54)(13,26,95,73)(14,27,96,74)(15,28,85,75)(16,29,86,76)(17,30,87,77)(18,31,88,78)(19,32,89,79)(20,33,90,80)(21,34,91,81)(22,35,92,82)(23,36,93,83)(24,25,94,84), (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), (1,96,61,20)(2,19,62,95)(3,94,63,18)(4,17,64,93)(5,92,65,16)(6,15,66,91)(7,90,67,14)(8,13,68,89)(9,88,69,24)(10,23,70,87)(11,86,71,22)(12,21,72,85)(25,45,78,51)(26,50,79,44)(27,43,80,49)(28,60,81,42)(29,41,82,59)(30,58,83,40)(31,39,84,57)(32,56,73,38)(33,37,74,55)(34,54,75,48)(35,47,76,53)(36,52,77,46)>;
G:=Group( (1,78,7,84)(2,73,8,79)(3,80,9,74)(4,75,10,81)(5,82,11,76)(6,77,12,83)(13,38,19,44)(14,45,20,39)(15,40,21,46)(16,47,22,41)(17,42,23,48)(18,37,24,43)(25,67,31,61)(26,62,32,68)(27,69,33,63)(28,64,34,70)(29,71,35,65)(30,66,36,72)(49,94,55,88)(50,89,56,95)(51,96,57,90)(52,91,58,85)(53,86,59,92)(54,93,60,87), (1,43,67,55)(2,44,68,56)(3,45,69,57)(4,46,70,58)(5,47,71,59)(6,48,72,60)(7,37,61,49)(8,38,62,50)(9,39,63,51)(10,40,64,52)(11,41,65,53)(12,42,66,54)(13,26,95,73)(14,27,96,74)(15,28,85,75)(16,29,86,76)(17,30,87,77)(18,31,88,78)(19,32,89,79)(20,33,90,80)(21,34,91,81)(22,35,92,82)(23,36,93,83)(24,25,94,84), (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), (1,96,61,20)(2,19,62,95)(3,94,63,18)(4,17,64,93)(5,92,65,16)(6,15,66,91)(7,90,67,14)(8,13,68,89)(9,88,69,24)(10,23,70,87)(11,86,71,22)(12,21,72,85)(25,45,78,51)(26,50,79,44)(27,43,80,49)(28,60,81,42)(29,41,82,59)(30,58,83,40)(31,39,84,57)(32,56,73,38)(33,37,74,55)(34,54,75,48)(35,47,76,53)(36,52,77,46) );
G=PermutationGroup([(1,78,7,84),(2,73,8,79),(3,80,9,74),(4,75,10,81),(5,82,11,76),(6,77,12,83),(13,38,19,44),(14,45,20,39),(15,40,21,46),(16,47,22,41),(17,42,23,48),(18,37,24,43),(25,67,31,61),(26,62,32,68),(27,69,33,63),(28,64,34,70),(29,71,35,65),(30,66,36,72),(49,94,55,88),(50,89,56,95),(51,96,57,90),(52,91,58,85),(53,86,59,92),(54,93,60,87)], [(1,43,67,55),(2,44,68,56),(3,45,69,57),(4,46,70,58),(5,47,71,59),(6,48,72,60),(7,37,61,49),(8,38,62,50),(9,39,63,51),(10,40,64,52),(11,41,65,53),(12,42,66,54),(13,26,95,73),(14,27,96,74),(15,28,85,75),(16,29,86,76),(17,30,87,77),(18,31,88,78),(19,32,89,79),(20,33,90,80),(21,34,91,81),(22,35,92,82),(23,36,93,83),(24,25,94,84)], [(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)], [(1,96,61,20),(2,19,62,95),(3,94,63,18),(4,17,64,93),(5,92,65,16),(6,15,66,91),(7,90,67,14),(8,13,68,89),(9,88,69,24),(10,23,70,87),(11,86,71,22),(12,21,72,85),(25,45,78,51),(26,50,79,44),(27,43,80,49),(28,60,81,42),(29,41,82,59),(30,58,83,40),(31,39,84,57),(32,56,73,38),(33,37,74,55),(34,54,75,48),(35,47,76,53),(36,52,77,46)])
Matrix representation ►G ⊆ 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] >;
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 |
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