metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.166D6, C6.752+ 1+4, C4⋊1D4.8S3, (C4×Dic6)⋊50C2, (D4×Dic3)⋊33C2, (C2×D4).114D6, (C2×C6).257C24, C12.133(C4○D4), C2.79(D4⋊6D6), C4.17(D4⋊2S3), C23.12D6⋊25C2, (C2×C12).634C23, (C4×C12).202C22, (C6×D4).160C22, (C22×C6).71C23, C23.73(C22×S3), C23.23D6⋊26C2, C4⋊Dic3.380C22, C22.278(S3×C23), Dic3⋊C4.163C22, C3⋊5(C22.53C24), (C4×Dic3).154C22, (C2×Dic3).133C23, (C2×Dic6).300C22, C6.D4.71C22, (C22×Dic3).156C22, C6.95(C2×C4○D4), (C3×C4⋊1D4).6C2, C2.59(C2×D4⋊2S3), (C2×C4).595(C22×S3), SmallGroup(192,1272)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.166D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 544 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C4×D4, C4×Q8, C22.D4, C4.4D4, C4⋊1D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C2×Dic6, C22×Dic3, C6×D4, C22.53C24, C4×Dic6, D4×Dic3, C23.23D6, C23.12D6, C3×C4⋊1D4, C42.166D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, D4⋊2S3, S3×C23, C22.53C24, C2×D4⋊2S3, D4⋊6D6, C42.166D6
►On 96 points
Generators in S96
(1 18 15 4)(2 5 16 13)(3 14 17 6)(7 43 34 54)(8 49 35 44)(9 45 36 50)(10 51 31 46)(11 47 32 52)(12 53 33 48)(19 42 39 22)(20 23 40 37)(21 38 41 24)(25 78 95 58)(26 59 96 73)(27 74 91 60)(28 55 92 75)(29 76 93 56)(30 57 94 77)(61 64 80 83)(62 84 81 65)(63 66 82 79)(67 70 86 89)(68 90 87 71)(69 72 88 85)
(1 43 37 10)(2 11 38 44)(3 45 39 12)(4 7 40 46)(5 47 41 8)(6 9 42 48)(13 52 21 35)(14 36 22 53)(15 54 23 31)(16 32 24 49)(17 50 19 33)(18 34 20 51)(25 83 55 89)(26 90 56 84)(27 79 57 85)(28 86 58 80)(29 81 59 87)(30 88 60 82)(61 92 67 78)(62 73 68 93)(63 94 69 74)(64 75 70 95)(65 96 71 76)(66 77 72 91)
(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 88 37 82)(2 87 38 81)(3 86 39 80)(4 85 40 79)(5 90 41 84)(6 89 42 83)(7 91 46 77)(8 96 47 76)(9 95 48 75)(10 94 43 74)(11 93 44 73)(12 92 45 78)(13 71 21 65)(14 70 22 64)(15 69 23 63)(16 68 24 62)(17 67 19 61)(18 72 20 66)(25 53 55 36)(26 52 56 35)(27 51 57 34)(28 50 58 33)(29 49 59 32)(30 54 60 31)
G:=sub<Sym(96)| (1,18,15,4)(2,5,16,13)(3,14,17,6)(7,43,34,54)(8,49,35,44)(9,45,36,50)(10,51,31,46)(11,47,32,52)(12,53,33,48)(19,42,39,22)(20,23,40,37)(21,38,41,24)(25,78,95,58)(26,59,96,73)(27,74,91,60)(28,55,92,75)(29,76,93,56)(30,57,94,77)(61,64,80,83)(62,84,81,65)(63,66,82,79)(67,70,86,89)(68,90,87,71)(69,72,88,85), (1,43,37,10)(2,11,38,44)(3,45,39,12)(4,7,40,46)(5,47,41,8)(6,9,42,48)(13,52,21,35)(14,36,22,53)(15,54,23,31)(16,32,24,49)(17,50,19,33)(18,34,20,51)(25,83,55,89)(26,90,56,84)(27,79,57,85)(28,86,58,80)(29,81,59,87)(30,88,60,82)(61,92,67,78)(62,73,68,93)(63,94,69,74)(64,75,70,95)(65,96,71,76)(66,77,72,91), (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), (1,88,37,82)(2,87,38,81)(3,86,39,80)(4,85,40,79)(5,90,41,84)(6,89,42,83)(7,91,46,77)(8,96,47,76)(9,95,48,75)(10,94,43,74)(11,93,44,73)(12,92,45,78)(13,71,21,65)(14,70,22,64)(15,69,23,63)(16,68,24,62)(17,67,19,61)(18,72,20,66)(25,53,55,36)(26,52,56,35)(27,51,57,34)(28,50,58,33)(29,49,59,32)(30,54,60,31)>;
G:=Group( (1,18,15,4)(2,5,16,13)(3,14,17,6)(7,43,34,54)(8,49,35,44)(9,45,36,50)(10,51,31,46)(11,47,32,52)(12,53,33,48)(19,42,39,22)(20,23,40,37)(21,38,41,24)(25,78,95,58)(26,59,96,73)(27,74,91,60)(28,55,92,75)(29,76,93,56)(30,57,94,77)(61,64,80,83)(62,84,81,65)(63,66,82,79)(67,70,86,89)(68,90,87,71)(69,72,88,85), (1,43,37,10)(2,11,38,44)(3,45,39,12)(4,7,40,46)(5,47,41,8)(6,9,42,48)(13,52,21,35)(14,36,22,53)(15,54,23,31)(16,32,24,49)(17,50,19,33)(18,34,20,51)(25,83,55,89)(26,90,56,84)(27,79,57,85)(28,86,58,80)(29,81,59,87)(30,88,60,82)(61,92,67,78)(62,73,68,93)(63,94,69,74)(64,75,70,95)(65,96,71,76)(66,77,72,91), (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), (1,88,37,82)(2,87,38,81)(3,86,39,80)(4,85,40,79)(5,90,41,84)(6,89,42,83)(7,91,46,77)(8,96,47,76)(9,95,48,75)(10,94,43,74)(11,93,44,73)(12,92,45,78)(13,71,21,65)(14,70,22,64)(15,69,23,63)(16,68,24,62)(17,67,19,61)(18,72,20,66)(25,53,55,36)(26,52,56,35)(27,51,57,34)(28,50,58,33)(29,49,59,32)(30,54,60,31) );
G=PermutationGroup([[(1,18,15,4),(2,5,16,13),(3,14,17,6),(7,43,34,54),(8,49,35,44),(9,45,36,50),(10,51,31,46),(11,47,32,52),(12,53,33,48),(19,42,39,22),(20,23,40,37),(21,38,41,24),(25,78,95,58),(26,59,96,73),(27,74,91,60),(28,55,92,75),(29,76,93,56),(30,57,94,77),(61,64,80,83),(62,84,81,65),(63,66,82,79),(67,70,86,89),(68,90,87,71),(69,72,88,85)], [(1,43,37,10),(2,11,38,44),(3,45,39,12),(4,7,40,46),(5,47,41,8),(6,9,42,48),(13,52,21,35),(14,36,22,53),(15,54,23,31),(16,32,24,49),(17,50,19,33),(18,34,20,51),(25,83,55,89),(26,90,56,84),(27,79,57,85),(28,86,58,80),(29,81,59,87),(30,88,60,82),(61,92,67,78),(62,73,68,93),(63,94,69,74),(64,75,70,95),(65,96,71,76),(66,77,72,91)], [(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)], [(1,88,37,82),(2,87,38,81),(3,86,39,80),(4,85,40,79),(5,90,41,84),(6,89,42,83),(7,91,46,77),(8,96,47,76),(9,95,48,75),(10,94,43,74),(11,93,44,73),(12,92,45,78),(13,71,21,65),(14,70,22,64),(15,69,23,63),(16,68,24,62),(17,67,19,61),(18,72,20,66),(25,53,55,36),(26,52,56,35),(27,51,57,34),(28,50,58,33),(29,49,59,32),(30,54,60,31)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2+ 1+4 | D4⋊2S3 | D4⋊6D6 |
| kernel | C42.166D6 | C4×Dic6 | D4×Dic3 | C23.23D6 | C23.12D6 | C3×C4⋊1D4 | C4⋊1D4 | C42 | C2×D4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 6 | 8 | 1 | 4 | 2 |
Matrix representation of C42.166D6 ►in GL6(𝔽13)
| 12 | 1 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 1 | 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 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
G:=sub<GL(6,GF(13))| [12,11,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,11,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[8,0,0,0,0,0,5,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,1,0,0,0,0,0,12] >;
C42.166D6 in GAP, Magma, Sage, TeX
C_4^2._{166}D_6 % in TeX
G:=Group("C4^2.166D6"); // GroupNames label
G:=SmallGroup(192,1272);
// by ID
G=gap.SmallGroup(192,1272);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,219,1571,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations