metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.131D6, (C4×Q8)⋊17S3, (C4×D12)⋊40C2, (S3×C42)⋊7C2, C4⋊C4.298D6, (Q8×C12)⋊15C2, D6⋊3Q8⋊46C2, D6.3(C4○D4), (C2×Q8).203D6, C42⋊2S3⋊17C2, D6.D4⋊50C2, C12⋊D4.14C2, C4.48(C4○D12), (C2×C6).124C24, C4.Dic6⋊47C2, C12.340(C4○D4), C12.23D4⋊33C2, (C4×C12).176C22, (C2×C12).622C23, D6⋊C4.104C22, C4.60(Q8⋊3S3), (C6×Q8).224C22, (C2×D12).217C22, C4⋊Dic3.308C22, C22.145(S3×C23), Dic3⋊C4.156C22, (C22×S3).181C23, C3⋊5(C23.36C23), (C4×Dic3).295C22, (C2×Dic3).217C23, C4⋊C4⋊S3⋊51C2, C2.31(S3×C4○D4), C2.63(C2×C4○D12), C6.146(C2×C4○D4), (S3×C2×C4).296C22, C2.12(C2×Q8⋊3S3), (C3×C4⋊C4).352C22, (C2×C4).170(C22×S3), SmallGroup(192,1139)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.131D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 568 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C6×Q8, C23.36C23, S3×C42, C42⋊2S3, C4×D12, C4×D12, C4.Dic6, D6.D4, C12⋊D4, C4⋊C4⋊S3, D6⋊3Q8, C12.23D4, Q8×C12, C42.131D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C4○D12, Q8⋊3S3, S3×C23, C23.36C23, C2×C4○D12, C2×Q8⋊3S3, S3×C4○D4, C42.131D6
►On 96 points
(1 38 73 70)(2 71 74 39)(3 40 75 72)(4 61 76 41)(5 42 77 62)(6 63 78 43)(7 44 79 64)(8 65 80 45)(9 46 81 66)(10 67 82 47)(11 48 83 68)(12 69 84 37)(13 88 60 32)(14 33 49 89)(15 90 50 34)(16 35 51 91)(17 92 52 36)(18 25 53 93)(19 94 54 26)(20 27 55 95)(21 96 56 28)(22 29 57 85)(23 86 58 30)(24 31 59 87)
(1 91 79 29)(2 92 80 30)(3 93 81 31)(4 94 82 32)(5 95 83 33)(6 96 84 34)(7 85 73 35)(8 86 74 36)(9 87 75 25)(10 88 76 26)(11 89 77 27)(12 90 78 28)(13 61 54 47)(14 62 55 48)(15 63 56 37)(16 64 57 38)(17 65 58 39)(18 66 59 40)(19 67 60 41)(20 68 49 42)(21 69 50 43)(22 70 51 44)(23 71 52 45)(24 72 53 46)
(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 72 7 66)(2 65 8 71)(3 70 9 64)(4 63 10 69)(5 68 11 62)(6 61 12 67)(13 28 19 34)(14 33 20 27)(15 26 21 32)(16 31 22 25)(17 36 23 30)(18 29 24 35)(37 76 43 82)(38 81 44 75)(39 74 45 80)(40 79 46 73)(41 84 47 78)(42 77 48 83)(49 89 55 95)(50 94 56 88)(51 87 57 93)(52 92 58 86)(53 85 59 91)(54 90 60 96)
G:=sub<Sym(96)| (1,38,73,70)(2,71,74,39)(3,40,75,72)(4,61,76,41)(5,42,77,62)(6,63,78,43)(7,44,79,64)(8,65,80,45)(9,46,81,66)(10,67,82,47)(11,48,83,68)(12,69,84,37)(13,88,60,32)(14,33,49,89)(15,90,50,34)(16,35,51,91)(17,92,52,36)(18,25,53,93)(19,94,54,26)(20,27,55,95)(21,96,56,28)(22,29,57,85)(23,86,58,30)(24,31,59,87), (1,91,79,29)(2,92,80,30)(3,93,81,31)(4,94,82,32)(5,95,83,33)(6,96,84,34)(7,85,73,35)(8,86,74,36)(9,87,75,25)(10,88,76,26)(11,89,77,27)(12,90,78,28)(13,61,54,47)(14,62,55,48)(15,63,56,37)(16,64,57,38)(17,65,58,39)(18,66,59,40)(19,67,60,41)(20,68,49,42)(21,69,50,43)(22,70,51,44)(23,71,52,45)(24,72,53,46), (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), (1,72,7,66)(2,65,8,71)(3,70,9,64)(4,63,10,69)(5,68,11,62)(6,61,12,67)(13,28,19,34)(14,33,20,27)(15,26,21,32)(16,31,22,25)(17,36,23,30)(18,29,24,35)(37,76,43,82)(38,81,44,75)(39,74,45,80)(40,79,46,73)(41,84,47,78)(42,77,48,83)(49,89,55,95)(50,94,56,88)(51,87,57,93)(52,92,58,86)(53,85,59,91)(54,90,60,96)>;
G:=Group( (1,38,73,70)(2,71,74,39)(3,40,75,72)(4,61,76,41)(5,42,77,62)(6,63,78,43)(7,44,79,64)(8,65,80,45)(9,46,81,66)(10,67,82,47)(11,48,83,68)(12,69,84,37)(13,88,60,32)(14,33,49,89)(15,90,50,34)(16,35,51,91)(17,92,52,36)(18,25,53,93)(19,94,54,26)(20,27,55,95)(21,96,56,28)(22,29,57,85)(23,86,58,30)(24,31,59,87), (1,91,79,29)(2,92,80,30)(3,93,81,31)(4,94,82,32)(5,95,83,33)(6,96,84,34)(7,85,73,35)(8,86,74,36)(9,87,75,25)(10,88,76,26)(11,89,77,27)(12,90,78,28)(13,61,54,47)(14,62,55,48)(15,63,56,37)(16,64,57,38)(17,65,58,39)(18,66,59,40)(19,67,60,41)(20,68,49,42)(21,69,50,43)(22,70,51,44)(23,71,52,45)(24,72,53,46), (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), (1,72,7,66)(2,65,8,71)(3,70,9,64)(4,63,10,69)(5,68,11,62)(6,61,12,67)(13,28,19,34)(14,33,20,27)(15,26,21,32)(16,31,22,25)(17,36,23,30)(18,29,24,35)(37,76,43,82)(38,81,44,75)(39,74,45,80)(40,79,46,73)(41,84,47,78)(42,77,48,83)(49,89,55,95)(50,94,56,88)(51,87,57,93)(52,92,58,86)(53,85,59,91)(54,90,60,96) );
G=PermutationGroup([[(1,38,73,70),(2,71,74,39),(3,40,75,72),(4,61,76,41),(5,42,77,62),(6,63,78,43),(7,44,79,64),(8,65,80,45),(9,46,81,66),(10,67,82,47),(11,48,83,68),(12,69,84,37),(13,88,60,32),(14,33,49,89),(15,90,50,34),(16,35,51,91),(17,92,52,36),(18,25,53,93),(19,94,54,26),(20,27,55,95),(21,96,56,28),(22,29,57,85),(23,86,58,30),(24,31,59,87)], [(1,91,79,29),(2,92,80,30),(3,93,81,31),(4,94,82,32),(5,95,83,33),(6,96,84,34),(7,85,73,35),(8,86,74,36),(9,87,75,25),(10,88,76,26),(11,89,77,27),(12,90,78,28),(13,61,54,47),(14,62,55,48),(15,63,56,37),(16,64,57,38),(17,65,58,39),(18,66,59,40),(19,67,60,41),(20,68,49,42),(21,69,50,43),(22,70,51,44),(23,71,52,45),(24,72,53,46)], [(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)], [(1,72,7,66),(2,65,8,71),(3,70,9,64),(4,63,10,69),(5,68,11,62),(6,61,12,67),(13,28,19,34),(14,33,20,27),(15,26,21,32),(16,31,22,25),(17,36,23,30),(18,29,24,35),(37,76,43,82),(38,81,44,75),(39,74,45,80),(40,79,46,73),(41,84,47,78),(42,77,48,83),(49,89,55,95),(50,94,56,88),(51,87,57,93),(52,92,58,86),(53,85,59,91),(54,90,60,96)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 4S | 4T | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| 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 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | Q8⋊3S3 | S3×C4○D4 |
| kernel | C42.131D6 | S3×C42 | C42⋊2S3 | C4×D12 | C4.Dic6 | D6.D4 | C12⋊D4 | C4⋊C4⋊S3 | D6⋊3Q8 | C12.23D4 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | D6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 8 | 4 | 8 | 2 | 2 |
Matrix representation of C42.131D6 ►in GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 4 | 2 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 8 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 5 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 2 | 11 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,5,0,0,0,0,11,4,0,0,9,2],[12,0,0,0,0,12,0,0,0,0,8,0,0,0,0,8],[0,8,0,0,8,0,0,0,0,0,8,5,0,0,8,0],[8,0,0,0,0,8,0,0,0,0,2,2,0,0,4,11] >;
C42.131D6 in GAP, Magma, Sage, TeX
C_4^2._{131}D_6 % in TeX
G:=Group("C4^2.131D6"); // GroupNames label
G:=SmallGroup(192,1139);
// by ID
G=gap.SmallGroup(192,1139);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,794,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations