metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.141D6, C6.902- 1+4, C4.33(S3×D4), (C4×S3).12D4, D6.46(C2×D4), C12.62(C2×D4), C4.4D4⋊9S3, C12⋊2Q8⋊30C2, (C2×D4).172D6, (C2×Q8).160D6, C22⋊C4.35D6, C6.89(C22×D4), C42⋊2S3⋊20C2, C23.9D6⋊41C2, (C2×C12).80C23, (C2×C6).219C24, C2.51(Q8○D12), Dic3.51(C2×D4), Dic3⋊Q8⋊24C2, C23.12D6⋊24C2, (C4×C12).185C22, D6⋊C4.110C22, (C6×D4).154C22, C4⋊Dic3.51C22, (C22×C6).49C23, C23.51(C22×S3), (C6×Q8).126C22, Dic3.D4⋊40C2, C22.240(S3×C23), Dic3⋊C4.120C22, (C22×S3).214C23, C3⋊4(C23.38C23), (C4×Dic3).133C22, (C2×Dic6).177C22, (C2×Dic3).114C23, C6.D4.54C22, (C22×Dic3).142C22, (C2×S3×Q8)⋊10C2, C2.62(C2×S3×D4), (C2×D4⋊2S3).9C2, (C3×C4.4D4)⋊11C2, (S3×C2×C4).120C22, (C2×C4).194(C22×S3), (C2×C3⋊D4).59C22, (C3×C22⋊C4).64C22, SmallGroup(192,1234)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.141D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >
Subgroups: 656 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, D4⋊2S3, S3×Q8, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C23.38C23, C12⋊2Q8, C42⋊2S3, Dic3.D4, C23.9D6, C23.12D6, Dic3⋊Q8, C3×C4.4D4, C2×D4⋊2S3, C2×S3×Q8, C42.141D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2- 1+4, S3×D4, S3×C23, C23.38C23, C2×S3×D4, Q8○D12, C42.141D6
►On 96 points
(1 49 7 55)(2 56 8 50)(3 51 9 57)(4 58 10 52)(5 53 11 59)(6 60 12 54)(13 42 19 48)(14 37 20 43)(15 44 21 38)(16 39 22 45)(17 46 23 40)(18 41 24 47)(25 81 31 75)(26 76 32 82)(27 83 33 77)(28 78 34 84)(29 73 35 79)(30 80 36 74)(61 90 67 96)(62 85 68 91)(63 92 69 86)(64 87 70 93)(65 94 71 88)(66 89 72 95)
(1 13 93 28)(2 35 94 20)(3 15 95 30)(4 25 96 22)(5 17 85 32)(6 27 86 24)(7 19 87 34)(8 29 88 14)(9 21 89 36)(10 31 90 16)(11 23 91 26)(12 33 92 18)(37 50 73 65)(38 72 74 57)(39 52 75 67)(40 62 76 59)(41 54 77 69)(42 64 78 49)(43 56 79 71)(44 66 80 51)(45 58 81 61)(46 68 82 53)(47 60 83 63)(48 70 84 55)
(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 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 36 31 30)(26 29 32 35)(27 34 33 28)(37 40 43 46)(38 45 44 39)(41 48 47 42)(49 54 55 60)(50 59 56 53)(51 52 57 58)(61 66 67 72)(62 71 68 65)(63 64 69 70)(73 76 79 82)(74 81 80 75)(77 84 83 78)(85 94 91 88)(86 87 92 93)(89 90 95 96)
G:=sub<Sym(96)| (1,49,7,55)(2,56,8,50)(3,51,9,57)(4,58,10,52)(5,53,11,59)(6,60,12,54)(13,42,19,48)(14,37,20,43)(15,44,21,38)(16,39,22,45)(17,46,23,40)(18,41,24,47)(25,81,31,75)(26,76,32,82)(27,83,33,77)(28,78,34,84)(29,73,35,79)(30,80,36,74)(61,90,67,96)(62,85,68,91)(63,92,69,86)(64,87,70,93)(65,94,71,88)(66,89,72,95), (1,13,93,28)(2,35,94,20)(3,15,95,30)(4,25,96,22)(5,17,85,32)(6,27,86,24)(7,19,87,34)(8,29,88,14)(9,21,89,36)(10,31,90,16)(11,23,91,26)(12,33,92,18)(37,50,73,65)(38,72,74,57)(39,52,75,67)(40,62,76,59)(41,54,77,69)(42,64,78,49)(43,56,79,71)(44,66,80,51)(45,58,81,61)(46,68,82,53)(47,60,83,63)(48,70,84,55), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,36,31,30)(26,29,32,35)(27,34,33,28)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,54,55,60)(50,59,56,53)(51,52,57,58)(61,66,67,72)(62,71,68,65)(63,64,69,70)(73,76,79,82)(74,81,80,75)(77,84,83,78)(85,94,91,88)(86,87,92,93)(89,90,95,96)>;
G:=Group( (1,49,7,55)(2,56,8,50)(3,51,9,57)(4,58,10,52)(5,53,11,59)(6,60,12,54)(13,42,19,48)(14,37,20,43)(15,44,21,38)(16,39,22,45)(17,46,23,40)(18,41,24,47)(25,81,31,75)(26,76,32,82)(27,83,33,77)(28,78,34,84)(29,73,35,79)(30,80,36,74)(61,90,67,96)(62,85,68,91)(63,92,69,86)(64,87,70,93)(65,94,71,88)(66,89,72,95), (1,13,93,28)(2,35,94,20)(3,15,95,30)(4,25,96,22)(5,17,85,32)(6,27,86,24)(7,19,87,34)(8,29,88,14)(9,21,89,36)(10,31,90,16)(11,23,91,26)(12,33,92,18)(37,50,73,65)(38,72,74,57)(39,52,75,67)(40,62,76,59)(41,54,77,69)(42,64,78,49)(43,56,79,71)(44,66,80,51)(45,58,81,61)(46,68,82,53)(47,60,83,63)(48,70,84,55), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,36,31,30)(26,29,32,35)(27,34,33,28)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,54,55,60)(50,59,56,53)(51,52,57,58)(61,66,67,72)(62,71,68,65)(63,64,69,70)(73,76,79,82)(74,81,80,75)(77,84,83,78)(85,94,91,88)(86,87,92,93)(89,90,95,96) );
G=PermutationGroup([[(1,49,7,55),(2,56,8,50),(3,51,9,57),(4,58,10,52),(5,53,11,59),(6,60,12,54),(13,42,19,48),(14,37,20,43),(15,44,21,38),(16,39,22,45),(17,46,23,40),(18,41,24,47),(25,81,31,75),(26,76,32,82),(27,83,33,77),(28,78,34,84),(29,73,35,79),(30,80,36,74),(61,90,67,96),(62,85,68,91),(63,92,69,86),(64,87,70,93),(65,94,71,88),(66,89,72,95)], [(1,13,93,28),(2,35,94,20),(3,15,95,30),(4,25,96,22),(5,17,85,32),(6,27,86,24),(7,19,87,34),(8,29,88,14),(9,21,89,36),(10,31,90,16),(11,23,91,26),(12,33,92,18),(37,50,73,65),(38,72,74,57),(39,52,75,67),(40,62,76,59),(41,54,77,69),(42,64,78,49),(43,56,79,71),(44,66,80,51),(45,58,81,61),(46,68,82,53),(47,60,83,63),(48,70,84,55)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,36,31,30),(26,29,32,35),(27,34,33,28),(37,40,43,46),(38,45,44,39),(41,48,47,42),(49,54,55,60),(50,59,56,53),(51,52,57,58),(61,66,67,72),(62,71,68,65),(63,64,69,70),(73,76,79,82),(74,81,80,75),(77,84,83,78),(85,94,91,88),(86,87,92,93),(89,90,95,96)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | 2- 1+4 | S3×D4 | Q8○D12 |
| kernel | C42.141D6 | C12⋊2Q8 | C42⋊2S3 | Dic3.D4 | C23.9D6 | C23.12D6 | Dic3⋊Q8 | C3×C4.4D4 | C2×D4⋊2S3 | C2×S3×Q8 | C4.4D4 | C4×S3 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C42.141D6 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 | 9 | 2 |
| 0 | 0 | 0 | 0 | 4 | 9 | 1 | 0 |
| 0 | 0 | 0 | 0 | 8 | 4 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 5 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 2 | 12 | 1 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 5 | 3 |
| 0 | 0 | 0 | 0 | 0 | 1 | 3 | 5 |
| 0 | 0 | 0 | 0 | 1 | 2 | 12 | 1 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 1 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,4,8,0,0,0,0,0,12,9,4,0,0,0,0,2,9,1,0,0,0,0,0,2,2,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,6,10,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,6,3],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,1,2,1,0,0,0,0,12,0,1,2,0,0,0,0,5,5,0,12,0,0,0,0,3,5,1,1],[1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0,1,2,0,0,0,0,12,1,2,1,0,0,0,0,5,3,12,0,0,0,0,0,3,5,1,1] >;
C42.141D6 in GAP, Magma, Sage, TeX
C_4^2._{141}D_6 % in TeX
G:=Group("C4^2.141D6"); // GroupNames label
G:=SmallGroup(192,1234);
// by ID
G=gap.SmallGroup(192,1234);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,675,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations