metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.90D6, C6.922+ 1+4, C6.482- 1+4, (C2×C12)⋊5Q8, C12⋊Q8⋊11C2, (C2×C4)⋊4Dic6, C4⋊C4.268D6, C12⋊2Q8⋊6C2, C12.78(C2×Q8), C2.6(D4○D12), (C2×C6).63C24, (C4×C12).7C22, C2.7(Q8○D12), C22⋊C4.91D6, C12.6Q8⋊4C2, C4.34(C2×Dic6), C6.11(C22×Q8), C4.Dic6⋊11C2, (C22×C4).203D6, C22.7(C2×Dic6), C42⋊C2.13S3, (C2×C12).142C23, Dic3⋊C4.2C22, C4⋊Dic3.32C22, C22.96(S3×C23), C2.13(C22×Dic6), C23.161(C22×S3), (C22×C6).133C23, Dic3.D4.1C2, (C2×Dic3).22C23, (C2×Dic6).23C22, (C4×Dic3).68C22, (C22×C12).223C22, C3⋊2(C23.41C23), C6.D4.92C22, C23.26D6.23C2, (C22×Dic3).85C22, (C2×C6).13(C2×Q8), (C2×C4⋊Dic3).44C2, (C3×C4⋊C4).304C22, (C2×C4).148(C22×S3), (C3×C42⋊C2).14C2, (C3×C22⋊C4).99C22, SmallGroup(192,1078)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.90D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 456 in 206 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, 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×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×C12, C23.41C23, C12⋊2Q8, C12.6Q8, Dic3.D4, C12⋊Q8, C4.Dic6, C2×C4⋊Dic3, C23.26D6, C3×C42⋊C2, C42.90D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, Dic6, C22×S3, C22×Q8, 2+ 1+4, 2- 1+4, C2×Dic6, S3×C23, C23.41C23, C22×Dic6, D4○D12, Q8○D12, C42.90D6
►On 96 points
(1 53 10 63)(2 51 11 61)(3 49 12 65)(4 50 7 66)(5 54 8 64)(6 52 9 62)(13 85 17 70)(14 89 18 68)(15 87 16 72)(19 58 24 84)(20 56 22 82)(21 60 23 80)(25 73 34 92)(26 77 35 96)(27 75 36 94)(28 59 46 79)(29 57 47 83)(30 55 48 81)(31 95 45 76)(32 93 43 74)(33 91 44 78)(37 86 41 71)(38 90 42 69)(39 88 40 67)
(1 45 4 25)(2 43 5 26)(3 44 6 27)(7 34 10 31)(8 35 11 32)(9 36 12 33)(13 48 39 24)(14 46 37 22)(15 47 38 23)(16 29 42 21)(17 30 40 19)(18 28 41 20)(49 78 52 75)(50 73 53 76)(51 74 54 77)(55 67 58 70)(56 68 59 71)(57 69 60 72)(61 93 64 96)(62 94 65 91)(63 95 66 92)(79 86 82 89)(80 87 83 90)(81 88 84 85)
(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 24 7 30)(2 23 8 29)(3 22 9 28)(4 48 10 19)(5 47 11 21)(6 46 12 20)(13 31 40 25)(14 33 41 27)(15 32 42 26)(16 43 38 35)(17 45 39 34)(18 44 37 36)(49 56 62 79)(50 55 63 84)(51 60 64 83)(52 59 65 82)(53 58 66 81)(54 57 61 80)(67 92 85 76)(68 91 86 75)(69 96 87 74)(70 95 88 73)(71 94 89 78)(72 93 90 77)
G:=sub<Sym(96)| (1,53,10,63)(2,51,11,61)(3,49,12,65)(4,50,7,66)(5,54,8,64)(6,52,9,62)(13,85,17,70)(14,89,18,68)(15,87,16,72)(19,58,24,84)(20,56,22,82)(21,60,23,80)(25,73,34,92)(26,77,35,96)(27,75,36,94)(28,59,46,79)(29,57,47,83)(30,55,48,81)(31,95,45,76)(32,93,43,74)(33,91,44,78)(37,86,41,71)(38,90,42,69)(39,88,40,67), (1,45,4,25)(2,43,5,26)(3,44,6,27)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,48,39,24)(14,46,37,22)(15,47,38,23)(16,29,42,21)(17,30,40,19)(18,28,41,20)(49,78,52,75)(50,73,53,76)(51,74,54,77)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,93,64,96)(62,94,65,91)(63,95,66,92)(79,86,82,89)(80,87,83,90)(81,88,84,85), (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), (1,24,7,30)(2,23,8,29)(3,22,9,28)(4,48,10,19)(5,47,11,21)(6,46,12,20)(13,31,40,25)(14,33,41,27)(15,32,42,26)(16,43,38,35)(17,45,39,34)(18,44,37,36)(49,56,62,79)(50,55,63,84)(51,60,64,83)(52,59,65,82)(53,58,66,81)(54,57,61,80)(67,92,85,76)(68,91,86,75)(69,96,87,74)(70,95,88,73)(71,94,89,78)(72,93,90,77)>;
G:=Group( (1,53,10,63)(2,51,11,61)(3,49,12,65)(4,50,7,66)(5,54,8,64)(6,52,9,62)(13,85,17,70)(14,89,18,68)(15,87,16,72)(19,58,24,84)(20,56,22,82)(21,60,23,80)(25,73,34,92)(26,77,35,96)(27,75,36,94)(28,59,46,79)(29,57,47,83)(30,55,48,81)(31,95,45,76)(32,93,43,74)(33,91,44,78)(37,86,41,71)(38,90,42,69)(39,88,40,67), (1,45,4,25)(2,43,5,26)(3,44,6,27)(7,34,10,31)(8,35,11,32)(9,36,12,33)(13,48,39,24)(14,46,37,22)(15,47,38,23)(16,29,42,21)(17,30,40,19)(18,28,41,20)(49,78,52,75)(50,73,53,76)(51,74,54,77)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,93,64,96)(62,94,65,91)(63,95,66,92)(79,86,82,89)(80,87,83,90)(81,88,84,85), (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), (1,24,7,30)(2,23,8,29)(3,22,9,28)(4,48,10,19)(5,47,11,21)(6,46,12,20)(13,31,40,25)(14,33,41,27)(15,32,42,26)(16,43,38,35)(17,45,39,34)(18,44,37,36)(49,56,62,79)(50,55,63,84)(51,60,64,83)(52,59,65,82)(53,58,66,81)(54,57,61,80)(67,92,85,76)(68,91,86,75)(69,96,87,74)(70,95,88,73)(71,94,89,78)(72,93,90,77) );
G=PermutationGroup([[(1,53,10,63),(2,51,11,61),(3,49,12,65),(4,50,7,66),(5,54,8,64),(6,52,9,62),(13,85,17,70),(14,89,18,68),(15,87,16,72),(19,58,24,84),(20,56,22,82),(21,60,23,80),(25,73,34,92),(26,77,35,96),(27,75,36,94),(28,59,46,79),(29,57,47,83),(30,55,48,81),(31,95,45,76),(32,93,43,74),(33,91,44,78),(37,86,41,71),(38,90,42,69),(39,88,40,67)], [(1,45,4,25),(2,43,5,26),(3,44,6,27),(7,34,10,31),(8,35,11,32),(9,36,12,33),(13,48,39,24),(14,46,37,22),(15,47,38,23),(16,29,42,21),(17,30,40,19),(18,28,41,20),(49,78,52,75),(50,73,53,76),(51,74,54,77),(55,67,58,70),(56,68,59,71),(57,69,60,72),(61,93,64,96),(62,94,65,91),(63,95,66,92),(79,86,82,89),(80,87,83,90),(81,88,84,85)], [(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)], [(1,24,7,30),(2,23,8,29),(3,22,9,28),(4,48,10,19),(5,47,11,21),(6,46,12,20),(13,31,40,25),(14,33,41,27),(15,32,42,26),(16,43,38,35),(17,45,39,34),(18,44,37,36),(49,56,62,79),(50,55,63,84),(51,60,64,83),(52,59,65,82),(53,58,66,81),(54,57,61,80),(67,92,85,76),(68,91,86,75),(69,96,87,74),(70,95,88,73),(71,94,89,78),(72,93,90,77)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | D6 | Dic6 | 2+ 1+4 | 2- 1+4 | D4○D12 | Q8○D12 |
| kernel | C42.90D6 | C12⋊2Q8 | C12.6Q8 | Dic3.D4 | C12⋊Q8 | C4.Dic6 | C2×C4⋊Dic3 | C23.26D6 | C3×C42⋊C2 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C42.90D6 ►in GL6(𝔽13)
| 3 | 7 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 1 |
| 0 | 0 | 0 | 8 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 12 | 8 | 0 |
| 0 | 0 | 12 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 8 | 1 | 0 |
| 0 | 0 | 5 | 0 | 0 | 12 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 11 | 0 |
| 0 | 0 | 8 | 0 | 0 | 2 |
| 0 | 0 | 1 | 0 | 0 | 5 |
| 0 | 0 | 0 | 12 | 8 | 0 |
G:=sub<GL(6,GF(13))| [3,6,0,0,0,0,7,10,0,0,0,0,0,0,0,12,8,0,0,0,12,0,0,8,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,12,0,0,0,5,12,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,5,0,0,0,12,8,0,0,0,0,0,1,0,0,0,0,0,0,12],[8,0,0,0,0,0,5,5,0,0,0,0,0,0,0,8,1,0,0,0,5,0,0,12,0,0,11,0,0,8,0,0,0,2,5,0] >;
C42.90D6 in GAP, Magma, Sage, TeX
C_4^2._{90}D_6 % in TeX
G:=Group("C4^2.90D6"); // GroupNames label
G:=SmallGroup(192,1078);
// by ID
G=gap.SmallGroup(192,1078);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,758,184,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations