metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.113D6, C6.192+ 1+4, (C4×D4)⋊20S3, (C4×D12)⋊32C2, (D4×C12)⋊22C2, Dic3⋊D4⋊9C2, D6⋊3D4⋊9C2, 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), (C4×C12).157C22, (C2×C12).700C23, (C6×D4).262C22, (C2×D12).213C22, C23.28D6⋊17C2, (C22×S3).36C23, C4⋊Dic3.200C22, (C22×C6).172C23, C23.109(C22×S3), C22.127(S3×C23), (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
Generators and relations for C42.113D6
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 >
Subgroups: 600 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, C22.47C24, C12.6Q8, C4×D12, C23.8D6, Dic3⋊D4, S3×C4⋊C4, C4⋊C4⋊7S3, C4×C3⋊D4, C23.28D6, D6⋊3D4, D4×C12, C42.113D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, S3×C23, C22.47C24, C2×C4○D12, D4⋊6D6, S3×C4○D4, C42.113D6
►On 96 points
(1 28 70 48)(2 37 71 29)(3 30 72 38)(4 39 61 31)(5 32 62 40)(6 41 63 33)(7 34 64 42)(8 43 65 35)(9 36 66 44)(10 45 67 25)(11 26 68 46)(12 47 69 27)(13 54 87 83)(14 84 88 55)(15 56 89 73)(16 74 90 57)(17 58 91 75)(18 76 92 59)(19 60 93 77)(20 78 94 49)(21 50 95 79)(22 80 96 51)(23 52 85 81)(24 82 86 53)
(1 55 7 49)(2 56 8 50)(3 57 9 51)(4 58 10 52)(5 59 11 53)(6 60 12 54)(13 33 19 27)(14 34 20 28)(15 35 21 29)(16 36 22 30)(17 25 23 31)(18 26 24 32)(37 89 43 95)(38 90 44 96)(39 91 45 85)(40 92 46 86)(41 93 47 87)(42 94 48 88)(61 75 67 81)(62 76 68 82)(63 77 69 83)(64 78 70 84)(65 79 71 73)(66 80 72 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 87 7 93)(2 92 8 86)(3 85 9 91)(4 90 10 96)(5 95 11 89)(6 88 12 94)(13 64 19 70)(14 69 20 63)(15 62 21 68)(16 67 22 61)(17 72 23 66)(18 65 24 71)(25 51 31 57)(26 56 32 50)(27 49 33 55)(28 54 34 60)(29 59 35 53)(30 52 36 58)(37 76 43 82)(38 81 44 75)(39 74 45 80)(40 79 46 73)(41 84 47 78)(42 77 48 83)
G:=sub<Sym(96)| (1,28,70,48)(2,37,71,29)(3,30,72,38)(4,39,61,31)(5,32,62,40)(6,41,63,33)(7,34,64,42)(8,43,65,35)(9,36,66,44)(10,45,67,25)(11,26,68,46)(12,47,69,27)(13,54,87,83)(14,84,88,55)(15,56,89,73)(16,74,90,57)(17,58,91,75)(18,76,92,59)(19,60,93,77)(20,78,94,49)(21,50,95,79)(22,80,96,51)(23,52,85,81)(24,82,86,53), (1,55,7,49)(2,56,8,50)(3,57,9,51)(4,58,10,52)(5,59,11,53)(6,60,12,54)(13,33,19,27)(14,34,20,28)(15,35,21,29)(16,36,22,30)(17,25,23,31)(18,26,24,32)(37,89,43,95)(38,90,44,96)(39,91,45,85)(40,92,46,86)(41,93,47,87)(42,94,48,88)(61,75,67,81)(62,76,68,82)(63,77,69,83)(64,78,70,84)(65,79,71,73)(66,80,72,74), (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), (1,87,7,93)(2,92,8,86)(3,85,9,91)(4,90,10,96)(5,95,11,89)(6,88,12,94)(13,64,19,70)(14,69,20,63)(15,62,21,68)(16,67,22,61)(17,72,23,66)(18,65,24,71)(25,51,31,57)(26,56,32,50)(27,49,33,55)(28,54,34,60)(29,59,35,53)(30,52,36,58)(37,76,43,82)(38,81,44,75)(39,74,45,80)(40,79,46,73)(41,84,47,78)(42,77,48,83)>;
G:=Group( (1,28,70,48)(2,37,71,29)(3,30,72,38)(4,39,61,31)(5,32,62,40)(6,41,63,33)(7,34,64,42)(8,43,65,35)(9,36,66,44)(10,45,67,25)(11,26,68,46)(12,47,69,27)(13,54,87,83)(14,84,88,55)(15,56,89,73)(16,74,90,57)(17,58,91,75)(18,76,92,59)(19,60,93,77)(20,78,94,49)(21,50,95,79)(22,80,96,51)(23,52,85,81)(24,82,86,53), (1,55,7,49)(2,56,8,50)(3,57,9,51)(4,58,10,52)(5,59,11,53)(6,60,12,54)(13,33,19,27)(14,34,20,28)(15,35,21,29)(16,36,22,30)(17,25,23,31)(18,26,24,32)(37,89,43,95)(38,90,44,96)(39,91,45,85)(40,92,46,86)(41,93,47,87)(42,94,48,88)(61,75,67,81)(62,76,68,82)(63,77,69,83)(64,78,70,84)(65,79,71,73)(66,80,72,74), (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), (1,87,7,93)(2,92,8,86)(3,85,9,91)(4,90,10,96)(5,95,11,89)(6,88,12,94)(13,64,19,70)(14,69,20,63)(15,62,21,68)(16,67,22,61)(17,72,23,66)(18,65,24,71)(25,51,31,57)(26,56,32,50)(27,49,33,55)(28,54,34,60)(29,59,35,53)(30,52,36,58)(37,76,43,82)(38,81,44,75)(39,74,45,80)(40,79,46,73)(41,84,47,78)(42,77,48,83) );
G=PermutationGroup([[(1,28,70,48),(2,37,71,29),(3,30,72,38),(4,39,61,31),(5,32,62,40),(6,41,63,33),(7,34,64,42),(8,43,65,35),(9,36,66,44),(10,45,67,25),(11,26,68,46),(12,47,69,27),(13,54,87,83),(14,84,88,55),(15,56,89,73),(16,74,90,57),(17,58,91,75),(18,76,92,59),(19,60,93,77),(20,78,94,49),(21,50,95,79),(22,80,96,51),(23,52,85,81),(24,82,86,53)], [(1,55,7,49),(2,56,8,50),(3,57,9,51),(4,58,10,52),(5,59,11,53),(6,60,12,54),(13,33,19,27),(14,34,20,28),(15,35,21,29),(16,36,22,30),(17,25,23,31),(18,26,24,32),(37,89,43,95),(38,90,44,96),(39,91,45,85),(40,92,46,86),(41,93,47,87),(42,94,48,88),(61,75,67,81),(62,76,68,82),(63,77,69,83),(64,78,70,84),(65,79,71,73),(66,80,72,74)], [(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)], [(1,87,7,93),(2,92,8,86),(3,85,9,91),(4,90,10,96),(5,95,11,89),(6,88,12,94),(13,64,19,70),(14,69,20,63),(15,62,21,68),(16,67,22,61),(17,72,23,66),(18,65,24,71),(25,51,31,57),(26,56,32,50),(27,49,33,55),(28,54,34,60),(29,59,35,53),(30,52,36,58),(37,76,43,82),(38,81,44,75),(39,74,45,80),(40,79,46,73),(41,84,47,78),(42,77,48,83)]])
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 |
Matrix representation of C42.113D6 ►in 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] >;
C42.113D6 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