metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.105D6, C6.562- 1+4, C4⋊C4.280D6, (C4×D4).12S3, (C4×Dic6)⋊27C2, (C2×D4).209D6, (D4×C12).13C2, Dic3.Q8⋊7C2, (C2×C6).85C24, C12.48D4⋊8C2, C22⋊C4.130D6, C2.14(Q8○D12), C12.6Q8⋊15C2, C23.8D6⋊6C2, (C22×C4).220D6, C12.291(C4○D4), (C2×C12).585C23, (C4×C12).147C22, (C6×D4).303C22, C23.26D6⋊6C2, C4.115(D4⋊2S3), C22.11(C4○D12), C23.16D6⋊27C2, C4⋊Dic3.296C22, C23.175(C22×S3), C22.113(S3×C23), (C22×C12).79C22, (C22×C6).155C23, (C2×Dic3).35C23, C6.D4.9C22, C23.23D6.5C2, Dic3⋊C4.153C22, C3⋊4(C22.46C24), (C4×Dic3).202C22, (C2×Dic6).236C22, (C22×Dic3).93C22, C6.37(C2×C4○D4), (C2×C4⋊Dic3)⋊23C2, C2.41(C2×C4○D12), (C2×C6).15(C4○D4), C2.19(C2×D4⋊2S3), (C3×C4⋊C4).321C22, (C2×C4).155(C22×S3), (C3×C22⋊C4).142C22, SmallGroup(192,1100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.105D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 440 in 214 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×C12, C6×D4, C22.46C24, C4×Dic6, C12.6Q8, C23.16D6, C23.8D6, Dic3.Q8, C12.48D4, C2×C4⋊Dic3, C23.26D6, C23.23D6, D4×C12, C42.105D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, D4⋊2S3, S3×C23, C22.46C24, C2×C4○D12, C2×D4⋊2S3, Q8○D12, C42.105D6
►On 96 points
(1 40 37 4)(2 5 38 41)(3 42 39 6)(7 51 54 10)(8 11 49 52)(9 53 50 12)(13 48 45 16)(14 17 46 43)(15 44 47 18)(19 36 33 22)(20 23 34 31)(21 32 35 24)(25 61 74 92)(26 93 75 62)(27 63 76 94)(28 95 77 64)(29 65 78 96)(30 91 73 66)(55 83 86 70)(56 71 87 84)(57 79 88 72)(58 67 89 80)(59 81 90 68)(60 69 85 82)
(1 7 31 17)(2 8 32 18)(3 9 33 13)(4 10 34 14)(5 11 35 15)(6 12 36 16)(19 45 39 50)(20 46 40 51)(21 47 41 52)(22 48 42 53)(23 43 37 54)(24 44 38 49)(25 89 95 70)(26 90 96 71)(27 85 91 72)(28 86 92 67)(29 87 93 68)(30 88 94 69)(55 61 80 77)(56 62 81 78)(57 63 82 73)(58 64 83 74)(59 65 84 75)(60 66 79 76)
(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 57)(2 87 38 56)(3 86 39 55)(4 85 40 60)(5 90 41 59)(6 89 42 58)(7 30 54 73)(8 29 49 78)(9 28 50 77)(10 27 51 76)(11 26 52 75)(12 25 53 74)(13 92 45 61)(14 91 46 66)(15 96 47 65)(16 95 48 64)(17 94 43 63)(18 93 44 62)(19 80 33 67)(20 79 34 72)(21 84 35 71)(22 83 36 70)(23 82 31 69)(24 81 32 68)
G:=sub<Sym(96)| (1,40,37,4)(2,5,38,41)(3,42,39,6)(7,51,54,10)(8,11,49,52)(9,53,50,12)(13,48,45,16)(14,17,46,43)(15,44,47,18)(19,36,33,22)(20,23,34,31)(21,32,35,24)(25,61,74,92)(26,93,75,62)(27,63,76,94)(28,95,77,64)(29,65,78,96)(30,91,73,66)(55,83,86,70)(56,71,87,84)(57,79,88,72)(58,67,89,80)(59,81,90,68)(60,69,85,82), (1,7,31,17)(2,8,32,18)(3,9,33,13)(4,10,34,14)(5,11,35,15)(6,12,36,16)(19,45,39,50)(20,46,40,51)(21,47,41,52)(22,48,42,53)(23,43,37,54)(24,44,38,49)(25,89,95,70)(26,90,96,71)(27,85,91,72)(28,86,92,67)(29,87,93,68)(30,88,94,69)(55,61,80,77)(56,62,81,78)(57,63,82,73)(58,64,83,74)(59,65,84,75)(60,66,79,76), (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), (1,88,37,57)(2,87,38,56)(3,86,39,55)(4,85,40,60)(5,90,41,59)(6,89,42,58)(7,30,54,73)(8,29,49,78)(9,28,50,77)(10,27,51,76)(11,26,52,75)(12,25,53,74)(13,92,45,61)(14,91,46,66)(15,96,47,65)(16,95,48,64)(17,94,43,63)(18,93,44,62)(19,80,33,67)(20,79,34,72)(21,84,35,71)(22,83,36,70)(23,82,31,69)(24,81,32,68)>;
G:=Group( (1,40,37,4)(2,5,38,41)(3,42,39,6)(7,51,54,10)(8,11,49,52)(9,53,50,12)(13,48,45,16)(14,17,46,43)(15,44,47,18)(19,36,33,22)(20,23,34,31)(21,32,35,24)(25,61,74,92)(26,93,75,62)(27,63,76,94)(28,95,77,64)(29,65,78,96)(30,91,73,66)(55,83,86,70)(56,71,87,84)(57,79,88,72)(58,67,89,80)(59,81,90,68)(60,69,85,82), (1,7,31,17)(2,8,32,18)(3,9,33,13)(4,10,34,14)(5,11,35,15)(6,12,36,16)(19,45,39,50)(20,46,40,51)(21,47,41,52)(22,48,42,53)(23,43,37,54)(24,44,38,49)(25,89,95,70)(26,90,96,71)(27,85,91,72)(28,86,92,67)(29,87,93,68)(30,88,94,69)(55,61,80,77)(56,62,81,78)(57,63,82,73)(58,64,83,74)(59,65,84,75)(60,66,79,76), (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), (1,88,37,57)(2,87,38,56)(3,86,39,55)(4,85,40,60)(5,90,41,59)(6,89,42,58)(7,30,54,73)(8,29,49,78)(9,28,50,77)(10,27,51,76)(11,26,52,75)(12,25,53,74)(13,92,45,61)(14,91,46,66)(15,96,47,65)(16,95,48,64)(17,94,43,63)(18,93,44,62)(19,80,33,67)(20,79,34,72)(21,84,35,71)(22,83,36,70)(23,82,31,69)(24,81,32,68) );
G=PermutationGroup([[(1,40,37,4),(2,5,38,41),(3,42,39,6),(7,51,54,10),(8,11,49,52),(9,53,50,12),(13,48,45,16),(14,17,46,43),(15,44,47,18),(19,36,33,22),(20,23,34,31),(21,32,35,24),(25,61,74,92),(26,93,75,62),(27,63,76,94),(28,95,77,64),(29,65,78,96),(30,91,73,66),(55,83,86,70),(56,71,87,84),(57,79,88,72),(58,67,89,80),(59,81,90,68),(60,69,85,82)], [(1,7,31,17),(2,8,32,18),(3,9,33,13),(4,10,34,14),(5,11,35,15),(6,12,36,16),(19,45,39,50),(20,46,40,51),(21,47,41,52),(22,48,42,53),(23,43,37,54),(24,44,38,49),(25,89,95,70),(26,90,96,71),(27,85,91,72),(28,86,92,67),(29,87,93,68),(30,88,94,69),(55,61,80,77),(56,62,81,78),(57,63,82,73),(58,64,83,74),(59,65,84,75),(60,66,79,76)], [(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)], [(1,88,37,57),(2,87,38,56),(3,86,39,55),(4,85,40,60),(5,90,41,59),(6,89,42,58),(7,30,54,73),(8,29,49,78),(9,28,50,77),(10,27,51,76),(11,26,52,75),(12,25,53,74),(13,92,45,61),(14,91,46,66),(15,96,47,65),(16,95,48,64),(17,94,43,63),(18,93,44,62),(19,80,33,67),(20,79,34,72),(21,84,35,71),(22,83,36,70),(23,82,31,69),(24,81,32,68)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 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 | 2 | 2 | 4 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 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⋊2S3 | Q8○D12 |
| kernel | C42.105D6 | C4×Dic6 | C12.6Q8 | C23.16D6 | C23.8D6 | Dic3.Q8 | C12.48D4 | C2×C4⋊Dic3 | C23.26D6 | C23.23D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C2×C6 | C22 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C42.105D6 ►in GL4(𝔽13) generated by
| 1 | 10 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 8 | 2 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 9 | 4 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 9 | 0 | 0 |
| 7 | 9 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,10,12,0,0,0,0,0,12,0,0,1,0],[8,0,0,0,2,5,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,4,3,0,0,0,0,0,1,0,0,1,0],[4,7,0,0,9,9,0,0,0,0,0,8,0,0,8,0] >;
C42.105D6 in GAP, Magma, Sage, TeX
C_4^2._{105}D_6 % in TeX
G:=Group("C4^2.105D6"); // GroupNames label
G:=SmallGroup(192,1100);
// by ID
G=gap.SmallGroup(192,1100);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations