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, C12⋊D4.14C2, D6.D4⋊50C2, C4.48(C4○D12), (C2×C6).124C24, C12.3Q8⋊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), (C2×Dic3).217C23, (C4×Dic3).295C22, 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
Subgroups: 568 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×10], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×3], Dic3 [×5], C12 [×4], C12 [×5], D6 [×2], D6 [×8], C2×C6, C42, C42 [×2], C42 [×3], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4 [×5], C2×D4 [×3], C2×Q8, C4×S3 [×10], D12 [×6], C2×Dic3 [×3], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3, C22×S3 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×3], Dic3⋊C4 [×4], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×10], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], S3×C2×C4 [×3], S3×C2×C4 [×2], C2×D12, C2×D12 [×2], C6×Q8, C23.36C23, S3×C42, C42⋊2S3 [×2], C4×D12, C4×D12 [×2], C12.3Q8, D6.D4 [×2], C12⋊D4, C4⋊C4⋊S3 [×2], D6⋊3Q8, C12.23D4, Q8×C12, C42.131D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], C4○D12 [×2], Q8⋊3S3 [×2], S3×C23, C23.36C23, C2×C4○D12, C2×Q8⋊3S3, S3×C4○D4, C42.131D6
Generators and relations
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 >
►On 96 points
(1 33 43 24)(2 13 44 34)(3 35 45 14)(4 15 46 36)(5 25 47 16)(6 17 48 26)(7 27 37 18)(8 19 38 28)(9 29 39 20)(10 21 40 30)(11 31 41 22)(12 23 42 32)(49 82 63 91)(50 92 64 83)(51 84 65 93)(52 94 66 73)(53 74 67 95)(54 96 68 75)(55 76 69 85)(56 86 70 77)(57 78 71 87)(58 88 72 79)(59 80 61 89)(60 90 62 81)
(1 96 37 81)(2 85 38 82)(3 86 39 83)(4 87 40 84)(5 88 41 73)(6 89 42 74)(7 90 43 75)(8 91 44 76)(9 92 45 77)(10 93 46 78)(11 94 47 79)(12 95 48 80)(13 55 28 63)(14 56 29 64)(15 57 30 65)(16 58 31 66)(17 59 32 67)(18 60 33 68)(19 49 34 69)(20 50 35 70)(21 51 36 71)(22 52 25 72)(23 53 26 61)(24 54 27 62)
(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 14 7 20)(2 19 8 13)(3 24 9 18)(4 17 10 23)(5 22 11 16)(6 15 12 21)(25 47 31 41)(26 40 32 46)(27 45 33 39)(28 38 34 44)(29 43 35 37)(30 48 36 42)(49 91 55 85)(50 96 56 90)(51 89 57 95)(52 94 58 88)(53 87 59 93)(54 92 60 86)(61 84 67 78)(62 77 68 83)(63 82 69 76)(64 75 70 81)(65 80 71 74)(66 73 72 79)
G:=sub<Sym(96)| (1,33,43,24)(2,13,44,34)(3,35,45,14)(4,15,46,36)(5,25,47,16)(6,17,48,26)(7,27,37,18)(8,19,38,28)(9,29,39,20)(10,21,40,30)(11,31,41,22)(12,23,42,32)(49,82,63,91)(50,92,64,83)(51,84,65,93)(52,94,66,73)(53,74,67,95)(54,96,68,75)(55,76,69,85)(56,86,70,77)(57,78,71,87)(58,88,72,79)(59,80,61,89)(60,90,62,81), (1,96,37,81)(2,85,38,82)(3,86,39,83)(4,87,40,84)(5,88,41,73)(6,89,42,74)(7,90,43,75)(8,91,44,76)(9,92,45,77)(10,93,46,78)(11,94,47,79)(12,95,48,80)(13,55,28,63)(14,56,29,64)(15,57,30,65)(16,58,31,66)(17,59,32,67)(18,60,33,68)(19,49,34,69)(20,50,35,70)(21,51,36,71)(22,52,25,72)(23,53,26,61)(24,54,27,62), (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), (1,14,7,20)(2,19,8,13)(3,24,9,18)(4,17,10,23)(5,22,11,16)(6,15,12,21)(25,47,31,41)(26,40,32,46)(27,45,33,39)(28,38,34,44)(29,43,35,37)(30,48,36,42)(49,91,55,85)(50,96,56,90)(51,89,57,95)(52,94,58,88)(53,87,59,93)(54,92,60,86)(61,84,67,78)(62,77,68,83)(63,82,69,76)(64,75,70,81)(65,80,71,74)(66,73,72,79)>;
G:=Group( (1,33,43,24)(2,13,44,34)(3,35,45,14)(4,15,46,36)(5,25,47,16)(6,17,48,26)(7,27,37,18)(8,19,38,28)(9,29,39,20)(10,21,40,30)(11,31,41,22)(12,23,42,32)(49,82,63,91)(50,92,64,83)(51,84,65,93)(52,94,66,73)(53,74,67,95)(54,96,68,75)(55,76,69,85)(56,86,70,77)(57,78,71,87)(58,88,72,79)(59,80,61,89)(60,90,62,81), (1,96,37,81)(2,85,38,82)(3,86,39,83)(4,87,40,84)(5,88,41,73)(6,89,42,74)(7,90,43,75)(8,91,44,76)(9,92,45,77)(10,93,46,78)(11,94,47,79)(12,95,48,80)(13,55,28,63)(14,56,29,64)(15,57,30,65)(16,58,31,66)(17,59,32,67)(18,60,33,68)(19,49,34,69)(20,50,35,70)(21,51,36,71)(22,52,25,72)(23,53,26,61)(24,54,27,62), (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), (1,14,7,20)(2,19,8,13)(3,24,9,18)(4,17,10,23)(5,22,11,16)(6,15,12,21)(25,47,31,41)(26,40,32,46)(27,45,33,39)(28,38,34,44)(29,43,35,37)(30,48,36,42)(49,91,55,85)(50,96,56,90)(51,89,57,95)(52,94,58,88)(53,87,59,93)(54,92,60,86)(61,84,67,78)(62,77,68,83)(63,82,69,76)(64,75,70,81)(65,80,71,74)(66,73,72,79) );
G=PermutationGroup([(1,33,43,24),(2,13,44,34),(3,35,45,14),(4,15,46,36),(5,25,47,16),(6,17,48,26),(7,27,37,18),(8,19,38,28),(9,29,39,20),(10,21,40,30),(11,31,41,22),(12,23,42,32),(49,82,63,91),(50,92,64,83),(51,84,65,93),(52,94,66,73),(53,74,67,95),(54,96,68,75),(55,76,69,85),(56,86,70,77),(57,78,71,87),(58,88,72,79),(59,80,61,89),(60,90,62,81)], [(1,96,37,81),(2,85,38,82),(3,86,39,83),(4,87,40,84),(5,88,41,73),(6,89,42,74),(7,90,43,75),(8,91,44,76),(9,92,45,77),(10,93,46,78),(11,94,47,79),(12,95,48,80),(13,55,28,63),(14,56,29,64),(15,57,30,65),(16,58,31,66),(17,59,32,67),(18,60,33,68),(19,49,34,69),(20,50,35,70),(21,51,36,71),(22,52,25,72),(23,53,26,61),(24,54,27,62)], [(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)], [(1,14,7,20),(2,19,8,13),(3,24,9,18),(4,17,10,23),(5,22,11,16),(6,15,12,21),(25,47,31,41),(26,40,32,46),(27,45,33,39),(28,38,34,44),(29,43,35,37),(30,48,36,42),(49,91,55,85),(50,96,56,90),(51,89,57,95),(52,94,58,88),(53,87,59,93),(54,92,60,86),(61,84,67,78),(62,77,68,83),(63,82,69,76),(64,75,70,81),(65,80,71,74),(66,73,72,79)])
Matrix representation ►G ⊆ 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] >;
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 | C12.3Q8 | 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 |
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