metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.159D6, C6.982- (1+4), C12⋊Q8⋊39C2, C4⋊C4.116D6, C42⋊2C2.S3, (C4×Dic6)⋊13C2, C22⋊C4.39D6, C12.6Q8⋊8C2, Dic3.Q8⋊36C2, (C4×C12).31C22, (C2×C6).245C24, (C2×C12).93C23, C2.62(Q8○D12), C12.3Q8⋊38C2, Dic6⋊C4⋊39C2, C4⋊Dic3.53C22, C23.61(C22×S3), (C22×C6).59C23, C23.8D6.3C2, Dic3.14(C4○D4), C22.266(S3×C23), Dic3.D4.4C2, C23.16D6.3C2, Dic3⋊C4.126C22, C3⋊6(C22.35C24), (C2×Dic3).127C23, (C2×Dic6).253C22, (C4×Dic3).217C22, C6.D4.61C22, (C22×Dic3).148C22, C2.92(S3×C4○D4), C6.203(C2×C4○D4), (C3×C4⋊C4).200C22, (C2×C4).302(C22×S3), (C3×C42⋊2C2).1C2, (C3×C22⋊C4).70C22, SmallGroup(192,1260)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 400 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2, C3, C4 [×15], C22, C22 [×3], C6 [×3], C6, C2×C4 [×6], C2×C4 [×10], Q8 [×4], C23, Dic3 [×2], Dic3 [×7], C12 [×6], C2×C6, C2×C6 [×3], C42, C42 [×5], C22⋊C4 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C4⋊C4 [×17], C22×C4, C2×Q8 [×2], Dic6 [×4], C2×Dic3 [×8], C2×Dic3 [×2], C2×C12 [×6], C22×C6, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2, C42⋊2C2 [×3], C4⋊Q8, C4×Dic3 [×5], Dic3⋊C4 [×12], C4⋊Dic3 [×5], C6.D4 [×3], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×2], C22×Dic3, C22.35C24, C4×Dic6, C12.6Q8, C23.16D6, Dic3.D4 [×2], C23.8D6 [×3], Dic6⋊C4, C12⋊Q8, Dic3.Q8 [×3], C12.3Q8, C3×C42⋊2C2, C42.159D6
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], S3×C23, C22.35C24, S3×C4○D4, Q8○D12 [×2], C42.159D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 79 52 34)(2 74 53 29)(3 81 54 36)(4 76 55 31)(5 83 56 26)(6 78 57 33)(7 73 58 28)(8 80 59 35)(9 75 60 30)(10 82 49 25)(11 77 50 32)(12 84 51 27)(13 93 68 43)(14 88 69 38)(15 95 70 45)(16 90 71 40)(17 85 72 47)(18 92 61 42)(19 87 62 37)(20 94 63 44)(21 89 64 39)(22 96 65 46)(23 91 66 41)(24 86 67 48)
(1 76 7 82)(2 26 8 32)(3 78 9 84)(4 28 10 34)(5 80 11 74)(6 30 12 36)(13 40 19 46)(14 85 20 91)(15 42 21 48)(16 87 22 93)(17 44 23 38)(18 89 24 95)(25 52 31 58)(27 54 33 60)(29 56 35 50)(37 65 43 71)(39 67 45 61)(41 69 47 63)(49 79 55 73)(51 81 57 75)(53 83 59 77)(62 96 68 90)(64 86 70 92)(66 88 72 94)
(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 93 52 43)(2 86 53 48)(3 91 54 41)(4 96 55 46)(5 89 56 39)(6 94 57 44)(7 87 58 37)(8 92 59 42)(9 85 60 47)(10 90 49 40)(11 95 50 45)(12 88 51 38)(13 73 68 28)(14 78 69 33)(15 83 70 26)(16 76 71 31)(17 81 72 36)(18 74 61 29)(19 79 62 34)(20 84 63 27)(21 77 64 32)(22 82 65 25)(23 75 66 30)(24 80 67 35)
G:=sub<Sym(96)| (1,79,52,34)(2,74,53,29)(3,81,54,36)(4,76,55,31)(5,83,56,26)(6,78,57,33)(7,73,58,28)(8,80,59,35)(9,75,60,30)(10,82,49,25)(11,77,50,32)(12,84,51,27)(13,93,68,43)(14,88,69,38)(15,95,70,45)(16,90,71,40)(17,85,72,47)(18,92,61,42)(19,87,62,37)(20,94,63,44)(21,89,64,39)(22,96,65,46)(23,91,66,41)(24,86,67,48), (1,76,7,82)(2,26,8,32)(3,78,9,84)(4,28,10,34)(5,80,11,74)(6,30,12,36)(13,40,19,46)(14,85,20,91)(15,42,21,48)(16,87,22,93)(17,44,23,38)(18,89,24,95)(25,52,31,58)(27,54,33,60)(29,56,35,50)(37,65,43,71)(39,67,45,61)(41,69,47,63)(49,79,55,73)(51,81,57,75)(53,83,59,77)(62,96,68,90)(64,86,70,92)(66,88,72,94), (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), (1,93,52,43)(2,86,53,48)(3,91,54,41)(4,96,55,46)(5,89,56,39)(6,94,57,44)(7,87,58,37)(8,92,59,42)(9,85,60,47)(10,90,49,40)(11,95,50,45)(12,88,51,38)(13,73,68,28)(14,78,69,33)(15,83,70,26)(16,76,71,31)(17,81,72,36)(18,74,61,29)(19,79,62,34)(20,84,63,27)(21,77,64,32)(22,82,65,25)(23,75,66,30)(24,80,67,35)>;
G:=Group( (1,79,52,34)(2,74,53,29)(3,81,54,36)(4,76,55,31)(5,83,56,26)(6,78,57,33)(7,73,58,28)(8,80,59,35)(9,75,60,30)(10,82,49,25)(11,77,50,32)(12,84,51,27)(13,93,68,43)(14,88,69,38)(15,95,70,45)(16,90,71,40)(17,85,72,47)(18,92,61,42)(19,87,62,37)(20,94,63,44)(21,89,64,39)(22,96,65,46)(23,91,66,41)(24,86,67,48), (1,76,7,82)(2,26,8,32)(3,78,9,84)(4,28,10,34)(5,80,11,74)(6,30,12,36)(13,40,19,46)(14,85,20,91)(15,42,21,48)(16,87,22,93)(17,44,23,38)(18,89,24,95)(25,52,31,58)(27,54,33,60)(29,56,35,50)(37,65,43,71)(39,67,45,61)(41,69,47,63)(49,79,55,73)(51,81,57,75)(53,83,59,77)(62,96,68,90)(64,86,70,92)(66,88,72,94), (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), (1,93,52,43)(2,86,53,48)(3,91,54,41)(4,96,55,46)(5,89,56,39)(6,94,57,44)(7,87,58,37)(8,92,59,42)(9,85,60,47)(10,90,49,40)(11,95,50,45)(12,88,51,38)(13,73,68,28)(14,78,69,33)(15,83,70,26)(16,76,71,31)(17,81,72,36)(18,74,61,29)(19,79,62,34)(20,84,63,27)(21,77,64,32)(22,82,65,25)(23,75,66,30)(24,80,67,35) );
G=PermutationGroup([(1,79,52,34),(2,74,53,29),(3,81,54,36),(4,76,55,31),(5,83,56,26),(6,78,57,33),(7,73,58,28),(8,80,59,35),(9,75,60,30),(10,82,49,25),(11,77,50,32),(12,84,51,27),(13,93,68,43),(14,88,69,38),(15,95,70,45),(16,90,71,40),(17,85,72,47),(18,92,61,42),(19,87,62,37),(20,94,63,44),(21,89,64,39),(22,96,65,46),(23,91,66,41),(24,86,67,48)], [(1,76,7,82),(2,26,8,32),(3,78,9,84),(4,28,10,34),(5,80,11,74),(6,30,12,36),(13,40,19,46),(14,85,20,91),(15,42,21,48),(16,87,22,93),(17,44,23,38),(18,89,24,95),(25,52,31,58),(27,54,33,60),(29,56,35,50),(37,65,43,71),(39,67,45,61),(41,69,47,63),(49,79,55,73),(51,81,57,75),(53,83,59,77),(62,96,68,90),(64,86,70,92),(66,88,72,94)], [(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)], [(1,93,52,43),(2,86,53,48),(3,91,54,41),(4,96,55,46),(5,89,56,39),(6,94,57,44),(7,87,58,37),(8,92,59,42),(9,85,60,47),(10,90,49,40),(11,95,50,45),(12,88,51,38),(13,73,68,28),(14,78,69,33),(15,83,70,26),(16,76,71,31),(17,81,72,36),(18,74,61,29),(19,79,62,34),(20,84,63,27),(21,77,64,32),(22,82,65,25),(23,75,66,30),(24,80,67,35)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 | 12 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 4 | 3 | 9 |
| 0 | 0 | 0 | 0 | 9 | 9 | 9 | 10 |
G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,11,1,1,12,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,12,0,0,0,0,11,1,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,1,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,2,12,12,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,6,3,7,9,0,0,0,0,5,7,4,9,0,0,0,0,0,0,3,9,0,0,0,0,0,0,9,10] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4○D4 | 2- (1+4) | S3×C4○D4 | Q8○D12 |
| kernel | C42.159D6 | C4×Dic6 | C12.6Q8 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic6⋊C4 | C12⋊Q8 | Dic3.Q8 | C12.3Q8 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{159}D_6 % in TeX
G:=Group("C4^2.159D6"); // GroupNames label
G:=SmallGroup(192,1260);
// by ID
G=gap.SmallGroup(192,1260);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations