metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.182D6, D6.1M4(2), Dic3.3M4(2), C8⋊C4⋊7S3, D6⋊C8.15C2, (C2×C8).156D6, Dic3⋊C8⋊36C2, C2.9(S3×M4(2)), (S3×C42).14C2, C3⋊1(C42.6C4), (C4×Dic3).15C4, C6.16(C2×M4(2)), C4.129(C4○D12), C12.245(C4○D4), (C4×C12).227C22, (C2×C12).811C23, C42.S3⋊18C2, (C2×C24).311C22, C6.9(C42⋊C2), C2.12(C42⋊2S3), (C4×Dic3).267C22, (S3×C2×C4).15C4, (C3×C8⋊C4)⋊17C2, C22.98(S3×C2×C4), (C2×C4).127(C4×S3), (C2×C12).146(C2×C4), (C2×C3⋊C8).189C22, (S3×C2×C4).269C22, (C2×C6).66(C22×C4), (C22×S3).52(C2×C4), (C2×C4).753(C22×S3), (C2×Dic3).81(C2×C4), SmallGroup(192,264)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.182D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a2b, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b2c5 >
Subgroups: 248 in 110 conjugacy classes, 51 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C8⋊C4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, C42.6C4, C42.S3, Dic3⋊C8, D6⋊C8, C3×C8⋊C4, S3×C42, C42.182D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C2×M4(2), S3×C2×C4, C4○D12, C42.6C4, C42⋊2S3, S3×M4(2), C42.182D6
►On 96 points
Generators in S96
(1 87 35 58)(2 76 36 71)(3 89 37 60)(4 78 38 49)(5 91 39 62)(6 80 40 51)(7 93 41 64)(8 82 42 53)(9 95 43 66)(10 84 44 55)(11 73 45 68)(12 86 46 57)(13 75 47 70)(14 88 48 59)(15 77 25 72)(16 90 26 61)(17 79 27 50)(18 92 28 63)(19 81 29 52)(20 94 30 65)(21 83 31 54)(22 96 32 67)(23 85 33 56)(24 74 34 69)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 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 18 29 46 13 6 41 34)(2 45 30 5 14 33 42 17)(3 4 31 32 15 16 43 44)(7 24 35 28 19 12 47 40)(8 27 36 11 20 39 48 23)(9 10 37 38 21 22 25 26)(49 95 96 60 61 83 84 72)(50 59 73 82 62 71 85 94)(51 81 74 70 63 93 86 58)(52 69 75 92 64 57 87 80)(53 91 76 56 65 79 88 68)(54 55 77 78 66 67 89 90)
G:=sub<Sym(96)| (1,87,35,58)(2,76,36,71)(3,89,37,60)(4,78,38,49)(5,91,39,62)(6,80,40,51)(7,93,41,64)(8,82,42,53)(9,95,43,66)(10,84,44,55)(11,73,45,68)(12,86,46,57)(13,75,47,70)(14,88,48,59)(15,77,25,72)(16,90,26,61)(17,79,27,50)(18,92,28,63)(19,81,29,52)(20,94,30,65)(21,83,31,54)(22,96,32,67)(23,85,33,56)(24,74,34,69), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84), (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), (1,18,29,46,13,6,41,34)(2,45,30,5,14,33,42,17)(3,4,31,32,15,16,43,44)(7,24,35,28,19,12,47,40)(8,27,36,11,20,39,48,23)(9,10,37,38,21,22,25,26)(49,95,96,60,61,83,84,72)(50,59,73,82,62,71,85,94)(51,81,74,70,63,93,86,58)(52,69,75,92,64,57,87,80)(53,91,76,56,65,79,88,68)(54,55,77,78,66,67,89,90)>;
G:=Group( (1,87,35,58)(2,76,36,71)(3,89,37,60)(4,78,38,49)(5,91,39,62)(6,80,40,51)(7,93,41,64)(8,82,42,53)(9,95,43,66)(10,84,44,55)(11,73,45,68)(12,86,46,57)(13,75,47,70)(14,88,48,59)(15,77,25,72)(16,90,26,61)(17,79,27,50)(18,92,28,63)(19,81,29,52)(20,94,30,65)(21,83,31,54)(22,96,32,67)(23,85,33,56)(24,74,34,69), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84), (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), (1,18,29,46,13,6,41,34)(2,45,30,5,14,33,42,17)(3,4,31,32,15,16,43,44)(7,24,35,28,19,12,47,40)(8,27,36,11,20,39,48,23)(9,10,37,38,21,22,25,26)(49,95,96,60,61,83,84,72)(50,59,73,82,62,71,85,94)(51,81,74,70,63,93,86,58)(52,69,75,92,64,57,87,80)(53,91,76,56,65,79,88,68)(54,55,77,78,66,67,89,90) );
G=PermutationGroup([[(1,87,35,58),(2,76,36,71),(3,89,37,60),(4,78,38,49),(5,91,39,62),(6,80,40,51),(7,93,41,64),(8,82,42,53),(9,95,43,66),(10,84,44,55),(11,73,45,68),(12,86,46,57),(13,75,47,70),(14,88,48,59),(15,77,25,72),(16,90,26,61),(17,79,27,50),(18,92,28,63),(19,81,29,52),(20,94,30,65),(21,83,31,54),(22,96,32,67),(23,85,33,56),(24,74,34,69)], [(1,19,13,7),(2,20,14,8),(3,21,15,9),(4,22,16,10),(5,23,17,11),(6,24,18,12),(25,43,37,31),(26,44,38,32),(27,45,39,33),(28,46,40,34),(29,47,41,35),(30,48,42,36),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60),(73,91,85,79),(74,92,86,80),(75,93,87,81),(76,94,88,82),(77,95,89,83),(78,96,90,84)], [(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)], [(1,18,29,46,13,6,41,34),(2,45,30,5,14,33,42,17),(3,4,31,32,15,16,43,44),(7,24,35,28,19,12,47,40),(8,27,36,11,20,39,48,23),(9,10,37,38,21,22,25,26),(49,95,96,60,61,83,84,72),(50,59,73,82,62,71,85,94),(51,81,74,70,63,93,86,58),(52,69,75,92,64,57,87,80),(53,91,76,56,65,79,88,68),(54,55,77,78,66,67,89,90)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4○D4 | M4(2) | C4×S3 | C4○D12 | S3×M4(2) |
| kernel | C42.182D6 | C42.S3 | Dic3⋊C8 | D6⋊C8 | C3×C8⋊C4 | S3×C42 | C4×Dic3 | S3×C2×C4 | C8⋊C4 | C42 | C2×C8 | Dic3 | C12 | D6 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 4 |
Matrix representation of C42.182D6 ►in GL6(𝔽73)
| 17 | 53 | 0 | 0 | 0 | 0 |
| 29 | 56 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 27 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 27 | 27 |
G:=sub<GL(6,GF(73))| [17,29,0,0,0,0,53,56,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,27,0,0,0,0,1,0,0,0,0,0,0,0,0,27,0,0,0,0,1,0,0,0,0,0,0,0,27,46,0,0,0,0,27,0],[0,27,0,0,0,0,1,0,0,0,0,0,0,0,0,27,0,0,0,0,72,0,0,0,0,0,0,0,46,27,0,0,0,0,0,27] >;
C42.182D6 in GAP, Magma, Sage, TeX
C_4^2._{182}D_6 % in TeX
G:=Group("C4^2.182D6"); // GroupNames label
G:=SmallGroup(192,264);
// by ID
G=gap.SmallGroup(192,264);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,422,387,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^-1,d^2=a^2*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations