metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.113D6, C6.192+ (1+4), (C4×D4)⋊20S3, D6⋊3D4⋊9C2, (D4×C12)⋊22C2, Dic3⋊D4⋊9C2, (C4×D12)⋊32C2, C4⋊C4.318D6, (C2×D4).219D6, (C22×C4).63D6, D6.29(C4○D4), C4.65(C4○D12), (C2×C6).102C24, D6⋊C4.86C22, C22⋊C4.115D6, C12.6Q8⋊16C2, C23.8D6⋊8C2, C12.110(C4○D4), C2.20(D4⋊6D6), (C2×C12).700C23, (C4×C12).157C22, (C6×D4).262C22, (C2×D12).213C22, C23.28D6⋊17C2, (C22×S3).36C23, C4⋊Dic3.200C22, (C22×C6).172C23, C22.127(S3×C23), C23.109(C22×S3), (C2×Dic3).43C23, Dic3⋊C4.100C22, (C22×C12).364C22, C3⋊4(C22.47C24), (C4×Dic3).205C22, C6.D4.14C22, (S3×C4⋊C4)⋊16C2, (C4×C3⋊D4)⋊44C2, C4⋊C4⋊7S3⋊15C2, C2.25(S3×C4○D4), C6.142(C2×C4○D4), C2.51(C2×C4○D12), (S3×C2×C4).66C22, (C3×C4⋊C4).331C22, (C2×C4).285(C22×S3), (C2×C3⋊D4).17C22, (C3×C22⋊C4).126C22, SmallGroup(192,1117)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 600 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×2], C4 [×10], C22, C22 [×13], S3 [×3], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], Dic3 [×6], C12 [×2], C12 [×4], D6 [×2], D6 [×5], C2×C6, C2×C6 [×6], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], C4×S3 [×6], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×6], C2×C12 [×3], C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×3], D6⋊C4 [×2], D6⋊C4 [×2], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], S3×C2×C4 [×2], C2×D12, C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, C22.47C24, C12.6Q8, C4×D12, C23.8D6 [×2], Dic3⋊D4 [×2], S3×C4⋊C4, C4⋊C4⋊7S3, C4×C3⋊D4 [×2], C23.28D6 [×2], D6⋊3D4 [×2], D4×C12, C42.113D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), C4○D12 [×2], S3×C23, C22.47C24, C2×C4○D12, D4⋊6D6, S3×C4○D4, C42.113D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 90 50 82)(2 83 51 91)(3 92 52 84)(4 73 53 93)(5 94 54 74)(6 75 55 95)(7 96 56 76)(8 77 57 85)(9 86 58 78)(10 79 59 87)(11 88 60 80)(12 81 49 89)(13 25 37 65)(14 66 38 26)(15 27 39 67)(16 68 40 28)(17 29 41 69)(18 70 42 30)(19 31 43 71)(20 72 44 32)(21 33 45 61)(22 62 46 34)(23 35 47 63)(24 64 48 36)
(1 72 7 66)(2 61 8 67)(3 62 9 68)(4 63 10 69)(5 64 11 70)(6 65 12 71)(13 81 19 75)(14 82 20 76)(15 83 21 77)(16 84 22 78)(17 73 23 79)(18 74 24 80)(25 49 31 55)(26 50 32 56)(27 51 33 57)(28 52 34 58)(29 53 35 59)(30 54 36 60)(37 89 43 95)(38 90 44 96)(39 91 45 85)(40 92 46 86)(41 93 47 87)(42 94 48 88)
(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 19 7 13)(2 24 8 18)(3 17 9 23)(4 22 10 16)(5 15 11 21)(6 20 12 14)(25 82 31 76)(26 75 32 81)(27 80 33 74)(28 73 34 79)(29 78 35 84)(30 83 36 77)(37 50 43 56)(38 55 44 49)(39 60 45 54)(40 53 46 59)(41 58 47 52)(42 51 48 57)(61 94 67 88)(62 87 68 93)(63 92 69 86)(64 85 70 91)(65 90 71 96)(66 95 72 89)
G:=sub<Sym(96)| (1,90,50,82)(2,83,51,91)(3,92,52,84)(4,73,53,93)(5,94,54,74)(6,75,55,95)(7,96,56,76)(8,77,57,85)(9,86,58,78)(10,79,59,87)(11,88,60,80)(12,81,49,89)(13,25,37,65)(14,66,38,26)(15,27,39,67)(16,68,40,28)(17,29,41,69)(18,70,42,30)(19,31,43,71)(20,72,44,32)(21,33,45,61)(22,62,46,34)(23,35,47,63)(24,64,48,36), (1,72,7,66)(2,61,8,67)(3,62,9,68)(4,63,10,69)(5,64,11,70)(6,65,12,71)(13,81,19,75)(14,82,20,76)(15,83,21,77)(16,84,22,78)(17,73,23,79)(18,74,24,80)(25,49,31,55)(26,50,32,56)(27,51,33,57)(28,52,34,58)(29,53,35,59)(30,54,36,60)(37,89,43,95)(38,90,44,96)(39,91,45,85)(40,92,46,86)(41,93,47,87)(42,94,48,88), (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), (1,19,7,13)(2,24,8,18)(3,17,9,23)(4,22,10,16)(5,15,11,21)(6,20,12,14)(25,82,31,76)(26,75,32,81)(27,80,33,74)(28,73,34,79)(29,78,35,84)(30,83,36,77)(37,50,43,56)(38,55,44,49)(39,60,45,54)(40,53,46,59)(41,58,47,52)(42,51,48,57)(61,94,67,88)(62,87,68,93)(63,92,69,86)(64,85,70,91)(65,90,71,96)(66,95,72,89)>;
G:=Group( (1,90,50,82)(2,83,51,91)(3,92,52,84)(4,73,53,93)(5,94,54,74)(6,75,55,95)(7,96,56,76)(8,77,57,85)(9,86,58,78)(10,79,59,87)(11,88,60,80)(12,81,49,89)(13,25,37,65)(14,66,38,26)(15,27,39,67)(16,68,40,28)(17,29,41,69)(18,70,42,30)(19,31,43,71)(20,72,44,32)(21,33,45,61)(22,62,46,34)(23,35,47,63)(24,64,48,36), (1,72,7,66)(2,61,8,67)(3,62,9,68)(4,63,10,69)(5,64,11,70)(6,65,12,71)(13,81,19,75)(14,82,20,76)(15,83,21,77)(16,84,22,78)(17,73,23,79)(18,74,24,80)(25,49,31,55)(26,50,32,56)(27,51,33,57)(28,52,34,58)(29,53,35,59)(30,54,36,60)(37,89,43,95)(38,90,44,96)(39,91,45,85)(40,92,46,86)(41,93,47,87)(42,94,48,88), (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), (1,19,7,13)(2,24,8,18)(3,17,9,23)(4,22,10,16)(5,15,11,21)(6,20,12,14)(25,82,31,76)(26,75,32,81)(27,80,33,74)(28,73,34,79)(29,78,35,84)(30,83,36,77)(37,50,43,56)(38,55,44,49)(39,60,45,54)(40,53,46,59)(41,58,47,52)(42,51,48,57)(61,94,67,88)(62,87,68,93)(63,92,69,86)(64,85,70,91)(65,90,71,96)(66,95,72,89) );
G=PermutationGroup([(1,90,50,82),(2,83,51,91),(3,92,52,84),(4,73,53,93),(5,94,54,74),(6,75,55,95),(7,96,56,76),(8,77,57,85),(9,86,58,78),(10,79,59,87),(11,88,60,80),(12,81,49,89),(13,25,37,65),(14,66,38,26),(15,27,39,67),(16,68,40,28),(17,29,41,69),(18,70,42,30),(19,31,43,71),(20,72,44,32),(21,33,45,61),(22,62,46,34),(23,35,47,63),(24,64,48,36)], [(1,72,7,66),(2,61,8,67),(3,62,9,68),(4,63,10,69),(5,64,11,70),(6,65,12,71),(13,81,19,75),(14,82,20,76),(15,83,21,77),(16,84,22,78),(17,73,23,79),(18,74,24,80),(25,49,31,55),(26,50,32,56),(27,51,33,57),(28,52,34,58),(29,53,35,59),(30,54,36,60),(37,89,43,95),(38,90,44,96),(39,91,45,85),(40,92,46,86),(41,93,47,87),(42,94,48,88)], [(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)], [(1,19,7,13),(2,24,8,18),(3,17,9,23),(4,22,10,16),(5,15,11,21),(6,20,12,14),(25,82,31,76),(26,75,32,81),(27,80,33,74),(28,73,34,79),(29,78,35,84),(30,83,36,77),(37,50,43,56),(38,55,44,49),(39,60,45,54),(40,53,46,59),(41,58,47,52),(42,51,48,57),(61,94,67,88),(62,87,68,93),(63,92,69,86),(64,85,70,91),(65,90,71,96),(66,95,72,89)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 3 | 7 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 5 | 0 | 0 |
| 8 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 5 | 0 |
| 2 | 2 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8],[3,6,0,0,7,10,0,0,0,0,5,0,0,0,0,5],[0,8,0,0,5,5,0,0,0,0,0,5,0,0,5,0],[2,4,0,0,2,11,0,0,0,0,0,8,0,0,8,0] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4P | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | ··· | 2 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 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 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ (1+4) | D4⋊6D6 | S3×C4○D4 |
| kernel | C42.113D6 | C12.6Q8 | C4×D12 | C23.8D6 | Dic3⋊D4 | S3×C4⋊C4 | C4⋊C4⋊7S3 | C4×C3⋊D4 | C23.28D6 | D6⋊3D4 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | D6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{113}D_6 % in TeX
G:=Group("C4^2.113D6"); // GroupNames label
G:=SmallGroup(192,1117);
// by ID
G=gap.SmallGroup(192,1117);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,1571,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^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^-1,d*c*d^-1=c^5>;
// generators/relations