metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.189D6, C4⋊C4.212D6, (S3×C42)⋊20C2, D6⋊Q8⋊42C2, Dic3⋊D4.3C2, C22⋊C4.78D6, Dic3⋊5D4⋊41C2, C42⋊2C2⋊10S3, C42⋊3S3⋊14C2, D6.13(C4○D4), D6.D4⋊40C2, C23.9D6⋊50C2, Dic3.Q8⋊37C2, (C2×C12).95C23, (C2×C6).250C24, D6⋊C4.45C22, Dic6⋊C4⋊41C2, Dic3⋊4D4⋊35C2, (C4×C12).234C22, (C22×C6).64C23, C23.66(C22×S3), Dic3.15(C4○D4), C23.11D6⋊46C2, (C2×D12).168C22, C23.16D6⋊21C2, Dic3⋊C4.72C22, C4⋊Dic3.246C22, C22.271(S3×C23), (C22×S3).224C23, (C2×Dic6).183C22, (C4×Dic3).150C22, (C2×Dic3).313C23, C6.D4.66C22, C3⋊11(C23.36C23), (C22×Dic3).150C22, C4⋊C4⋊7S3⋊40C2, C2.97(S3×C4○D4), C6.208(C2×C4○D4), (C3×C42⋊2C2)⋊5C2, (S3×C2×C4).300C22, (C2×C4).87(C22×S3), (C3×C4⋊C4).202C22, (C2×C3⋊D4).70C22, (C3×C22⋊C4).75C22, SmallGroup(192,1265)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 560 in 234 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×14], C22, C22 [×10], S3 [×3], C6 [×3], C6, C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×6], D6 [×2], D6 [×5], C2×C6, C2×C6 [×3], C42, C42 [×5], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic6 [×2], C4×S3 [×8], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×6], C22×S3 [×2], C22×C6, C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2, C42⋊2C2, C4×Dic3 [×5], Dic3⋊C4 [×6], C4⋊Dic3, D6⋊C4 [×6], C6.D4, C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4 [×4], C2×D12, C22×Dic3, C2×C3⋊D4 [×2], C23.36C23, S3×C42, C42⋊3S3, C23.16D6, Dic3⋊4D4 [×2], C23.9D6, Dic3⋊D4, C23.11D6, Dic6⋊C4, Dic3.Q8, C4⋊C4⋊7S3, Dic3⋊5D4, D6.D4, D6⋊Q8, C3×C42⋊2C2, C42.189D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], S3×C23, C23.36C23, S3×C4○D4 [×3], C42.189D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >
►On 96 points
(1 64 19 83)(2 68 20 54)(3 66 21 79)(4 70 22 50)(5 62 23 81)(6 72 24 52)(7 84 27 65)(8 49 28 69)(9 80 29 61)(10 51 30 71)(11 82 25 63)(12 53 26 67)(13 38 90 47)(14 55 85 34)(15 40 86 43)(16 57 87 36)(17 42 88 45)(18 59 89 32)(31 78 58 93)(33 74 60 95)(35 76 56 91)(37 94 46 73)(39 96 48 75)(41 92 44 77)
(1 56 12 43)(2 36 7 41)(3 58 8 45)(4 32 9 37)(5 60 10 47)(6 34 11 39)(13 62 95 51)(14 82 96 72)(15 64 91 53)(16 84 92 68)(17 66 93 49)(18 80 94 70)(19 35 26 40)(20 57 27 44)(21 31 28 42)(22 59 29 46)(23 33 30 38)(24 55 25 48)(50 89 61 73)(52 85 63 75)(54 87 65 77)(67 86 83 76)(69 88 79 78)(71 90 81 74)
(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 78 19 93)(2 92 20 77)(3 76 21 91)(4 96 22 75)(5 74 23 95)(6 94 24 73)(7 16 27 87)(8 86 28 15)(9 14 29 85)(10 90 30 13)(11 18 25 89)(12 88 26 17)(31 67 58 53)(32 52 59 72)(33 71 60 51)(34 50 55 70)(35 69 56 49)(36 54 57 68)(37 63 46 82)(38 81 47 62)(39 61 48 80)(40 79 43 66)(41 65 44 84)(42 83 45 64)
G:=sub<Sym(96)| (1,64,19,83)(2,68,20,54)(3,66,21,79)(4,70,22,50)(5,62,23,81)(6,72,24,52)(7,84,27,65)(8,49,28,69)(9,80,29,61)(10,51,30,71)(11,82,25,63)(12,53,26,67)(13,38,90,47)(14,55,85,34)(15,40,86,43)(16,57,87,36)(17,42,88,45)(18,59,89,32)(31,78,58,93)(33,74,60,95)(35,76,56,91)(37,94,46,73)(39,96,48,75)(41,92,44,77), (1,56,12,43)(2,36,7,41)(3,58,8,45)(4,32,9,37)(5,60,10,47)(6,34,11,39)(13,62,95,51)(14,82,96,72)(15,64,91,53)(16,84,92,68)(17,66,93,49)(18,80,94,70)(19,35,26,40)(20,57,27,44)(21,31,28,42)(22,59,29,46)(23,33,30,38)(24,55,25,48)(50,89,61,73)(52,85,63,75)(54,87,65,77)(67,86,83,76)(69,88,79,78)(71,90,81,74), (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), (1,78,19,93)(2,92,20,77)(3,76,21,91)(4,96,22,75)(5,74,23,95)(6,94,24,73)(7,16,27,87)(8,86,28,15)(9,14,29,85)(10,90,30,13)(11,18,25,89)(12,88,26,17)(31,67,58,53)(32,52,59,72)(33,71,60,51)(34,50,55,70)(35,69,56,49)(36,54,57,68)(37,63,46,82)(38,81,47,62)(39,61,48,80)(40,79,43,66)(41,65,44,84)(42,83,45,64)>;
G:=Group( (1,64,19,83)(2,68,20,54)(3,66,21,79)(4,70,22,50)(5,62,23,81)(6,72,24,52)(7,84,27,65)(8,49,28,69)(9,80,29,61)(10,51,30,71)(11,82,25,63)(12,53,26,67)(13,38,90,47)(14,55,85,34)(15,40,86,43)(16,57,87,36)(17,42,88,45)(18,59,89,32)(31,78,58,93)(33,74,60,95)(35,76,56,91)(37,94,46,73)(39,96,48,75)(41,92,44,77), (1,56,12,43)(2,36,7,41)(3,58,8,45)(4,32,9,37)(5,60,10,47)(6,34,11,39)(13,62,95,51)(14,82,96,72)(15,64,91,53)(16,84,92,68)(17,66,93,49)(18,80,94,70)(19,35,26,40)(20,57,27,44)(21,31,28,42)(22,59,29,46)(23,33,30,38)(24,55,25,48)(50,89,61,73)(52,85,63,75)(54,87,65,77)(67,86,83,76)(69,88,79,78)(71,90,81,74), (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), (1,78,19,93)(2,92,20,77)(3,76,21,91)(4,96,22,75)(5,74,23,95)(6,94,24,73)(7,16,27,87)(8,86,28,15)(9,14,29,85)(10,90,30,13)(11,18,25,89)(12,88,26,17)(31,67,58,53)(32,52,59,72)(33,71,60,51)(34,50,55,70)(35,69,56,49)(36,54,57,68)(37,63,46,82)(38,81,47,62)(39,61,48,80)(40,79,43,66)(41,65,44,84)(42,83,45,64) );
G=PermutationGroup([(1,64,19,83),(2,68,20,54),(3,66,21,79),(4,70,22,50),(5,62,23,81),(6,72,24,52),(7,84,27,65),(8,49,28,69),(9,80,29,61),(10,51,30,71),(11,82,25,63),(12,53,26,67),(13,38,90,47),(14,55,85,34),(15,40,86,43),(16,57,87,36),(17,42,88,45),(18,59,89,32),(31,78,58,93),(33,74,60,95),(35,76,56,91),(37,94,46,73),(39,96,48,75),(41,92,44,77)], [(1,56,12,43),(2,36,7,41),(3,58,8,45),(4,32,9,37),(5,60,10,47),(6,34,11,39),(13,62,95,51),(14,82,96,72),(15,64,91,53),(16,84,92,68),(17,66,93,49),(18,80,94,70),(19,35,26,40),(20,57,27,44),(21,31,28,42),(22,59,29,46),(23,33,30,38),(24,55,25,48),(50,89,61,73),(52,85,63,75),(54,87,65,77),(67,86,83,76),(69,88,79,78),(71,90,81,74)], [(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)], [(1,78,19,93),(2,92,20,77),(3,76,21,91),(4,96,22,75),(5,74,23,95),(6,94,24,73),(7,16,27,87),(8,86,28,15),(9,14,29,85),(10,90,30,13),(11,18,25,89),(12,88,26,17),(31,67,58,53),(32,52,59,72),(33,71,60,51),(34,50,55,70),(35,69,56,49),(36,54,57,68),(37,63,46,82),(38,81,47,62),(39,61,48,80),(40,79,43,66),(41,65,44,84),(42,83,45,64)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 10 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,8,0,0,0,0,10,8],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,12],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,8,5,0,0,0,0,0,5] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 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 | 6 | 6 | 12 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | S3×C4○D4 |
| kernel | C42.189D6 | S3×C42 | C42⋊3S3 | C23.16D6 | Dic3⋊4D4 | C23.9D6 | Dic3⋊D4 | C23.11D6 | Dic6⋊C4 | Dic3.Q8 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | D6⋊Q8 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | D6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 8 | 4 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{189}D_6 % in TeX
G:=Group("C4^2.189D6"); // GroupNames label
G:=SmallGroup(192,1265);
// by ID
G=gap.SmallGroup(192,1265);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,794,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations