metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.118D6, C6.632- 1+4, C6.232+ 1+4, (C4×D4)⋊26S3, (D4×C12)⋊28C2, C4⋊C4.288D6, D6⋊Q8⋊9C2, (C2×D4).225D6, Dic3.Q8⋊8C2, (C22×C4).65D6, C42⋊3S3⋊11C2, C23.9D6⋊10C2, (C2×C6).108C24, D6⋊C4.67C22, C22⋊C4.120D6, C2.20(Q8○D12), C12.6Q8⋊26C2, C2.25(D4⋊6D6), C12.48D4⋊13C2, (C4×C12).220C22, (C2×C12).166C23, C23.14D6.4C2, (C6×D4).309C22, C22.7(C4○D12), C23.21D6⋊7C2, C23.28D6⋊5C2, C23.8D6⋊10C2, Dic3⋊C4.8C22, C4⋊Dic3.41C22, Dic3.D4⋊10C2, C23.23D6⋊19C2, (C22×S3).42C23, C23.115(C22×S3), C22.133(S3×C23), (C22×C6).178C23, (C2×Dic3).48C23, (C2×Dic6).29C22, (C22×C12).366C22, C3⋊2(C22.33C24), (C4×Dic3).207C22, C6.D4.17C22, (C22×Dic3).100C22, (C4×C3⋊D4)⋊47C2, C6.50(C2×C4○D4), C2.57(C2×C4○D12), (C2×Dic3⋊C4)⋊39C2, (C2×C6).18(C4○D4), (S3×C2×C4).204C22, (C3×C4⋊C4).336C22, (C2×C4).164(C22×S3), (C2×C3⋊D4).117C22, (C3×C22⋊C4).107C22, SmallGroup(192,1123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.118D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 520 in 218 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C22.33C24, C12.6Q8, C42⋊3S3, Dic3.D4, C23.8D6, C23.9D6, C23.21D6, Dic3.Q8, D6⋊Q8, C2×Dic3⋊C4, C12.48D4, C4×C3⋊D4, C23.28D6, C23.23D6, C23.14D6, D4×C12, C42.118D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, C4○D12, S3×C23, C22.33C24, C2×C4○D12, D4⋊6D6, Q8○D12, C42.118D6
►On 96 points
(1 82 73 14)(2 15 74 83)(3 84 75 16)(4 17 76 79)(5 80 77 18)(6 13 78 81)(7 70 51 44)(8 45 52 71)(9 72 53 46)(10 47 54 67)(11 68 49 48)(12 43 50 69)(19 28 87 96)(20 91 88 29)(21 30 89 92)(22 93 90 25)(23 26 85 94)(24 95 86 27)(31 66 57 40)(32 41 58 61)(33 62 59 42)(34 37 60 63)(35 64 55 38)(36 39 56 65)
(1 51 35 29)(2 8 36 92)(3 53 31 25)(4 10 32 94)(5 49 33 27)(6 12 34 96)(7 55 91 73)(9 57 93 75)(11 59 95 77)(13 43 37 19)(14 70 38 88)(15 45 39 21)(16 72 40 90)(17 47 41 23)(18 68 42 86)(20 82 44 64)(22 84 46 66)(24 80 48 62)(26 76 54 58)(28 78 50 60)(30 74 52 56)(61 85 79 67)(63 87 81 69)(65 89 83 71)
(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 6 73 78)(2 77 74 5)(3 4 75 76)(7 28 51 96)(8 95 52 27)(9 26 53 94)(10 93 54 25)(11 30 49 92)(12 91 50 29)(13 64 81 38)(14 37 82 63)(15 62 83 42)(16 41 84 61)(17 66 79 40)(18 39 80 65)(19 20 87 88)(21 24 89 86)(22 85 90 23)(31 32 57 58)(33 36 59 56)(34 55 60 35)(43 44 69 70)(45 48 71 68)(46 67 72 47)
G:=sub<Sym(96)| (1,82,73,14)(2,15,74,83)(3,84,75,16)(4,17,76,79)(5,80,77,18)(6,13,78,81)(7,70,51,44)(8,45,52,71)(9,72,53,46)(10,47,54,67)(11,68,49,48)(12,43,50,69)(19,28,87,96)(20,91,88,29)(21,30,89,92)(22,93,90,25)(23,26,85,94)(24,95,86,27)(31,66,57,40)(32,41,58,61)(33,62,59,42)(34,37,60,63)(35,64,55,38)(36,39,56,65), (1,51,35,29)(2,8,36,92)(3,53,31,25)(4,10,32,94)(5,49,33,27)(6,12,34,96)(7,55,91,73)(9,57,93,75)(11,59,95,77)(13,43,37,19)(14,70,38,88)(15,45,39,21)(16,72,40,90)(17,47,41,23)(18,68,42,86)(20,82,44,64)(22,84,46,66)(24,80,48,62)(26,76,54,58)(28,78,50,60)(30,74,52,56)(61,85,79,67)(63,87,81,69)(65,89,83,71), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,28,51,96)(8,95,52,27)(9,26,53,94)(10,93,54,25)(11,30,49,92)(12,91,50,29)(13,64,81,38)(14,37,82,63)(15,62,83,42)(16,41,84,61)(17,66,79,40)(18,39,80,65)(19,20,87,88)(21,24,89,86)(22,85,90,23)(31,32,57,58)(33,36,59,56)(34,55,60,35)(43,44,69,70)(45,48,71,68)(46,67,72,47)>;
G:=Group( (1,82,73,14)(2,15,74,83)(3,84,75,16)(4,17,76,79)(5,80,77,18)(6,13,78,81)(7,70,51,44)(8,45,52,71)(9,72,53,46)(10,47,54,67)(11,68,49,48)(12,43,50,69)(19,28,87,96)(20,91,88,29)(21,30,89,92)(22,93,90,25)(23,26,85,94)(24,95,86,27)(31,66,57,40)(32,41,58,61)(33,62,59,42)(34,37,60,63)(35,64,55,38)(36,39,56,65), (1,51,35,29)(2,8,36,92)(3,53,31,25)(4,10,32,94)(5,49,33,27)(6,12,34,96)(7,55,91,73)(9,57,93,75)(11,59,95,77)(13,43,37,19)(14,70,38,88)(15,45,39,21)(16,72,40,90)(17,47,41,23)(18,68,42,86)(20,82,44,64)(22,84,46,66)(24,80,48,62)(26,76,54,58)(28,78,50,60)(30,74,52,56)(61,85,79,67)(63,87,81,69)(65,89,83,71), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,28,51,96)(8,95,52,27)(9,26,53,94)(10,93,54,25)(11,30,49,92)(12,91,50,29)(13,64,81,38)(14,37,82,63)(15,62,83,42)(16,41,84,61)(17,66,79,40)(18,39,80,65)(19,20,87,88)(21,24,89,86)(22,85,90,23)(31,32,57,58)(33,36,59,56)(34,55,60,35)(43,44,69,70)(45,48,71,68)(46,67,72,47) );
G=PermutationGroup([[(1,82,73,14),(2,15,74,83),(3,84,75,16),(4,17,76,79),(5,80,77,18),(6,13,78,81),(7,70,51,44),(8,45,52,71),(9,72,53,46),(10,47,54,67),(11,68,49,48),(12,43,50,69),(19,28,87,96),(20,91,88,29),(21,30,89,92),(22,93,90,25),(23,26,85,94),(24,95,86,27),(31,66,57,40),(32,41,58,61),(33,62,59,42),(34,37,60,63),(35,64,55,38),(36,39,56,65)], [(1,51,35,29),(2,8,36,92),(3,53,31,25),(4,10,32,94),(5,49,33,27),(6,12,34,96),(7,55,91,73),(9,57,93,75),(11,59,95,77),(13,43,37,19),(14,70,38,88),(15,45,39,21),(16,72,40,90),(17,47,41,23),(18,68,42,86),(20,82,44,64),(22,84,46,66),(24,80,48,62),(26,76,54,58),(28,78,50,60),(30,74,52,56),(61,85,79,67),(63,87,81,69),(65,89,83,71)], [(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)], [(1,6,73,78),(2,77,74,5),(3,4,75,76),(7,28,51,96),(8,95,52,27),(9,26,53,94),(10,93,54,25),(11,30,49,92),(12,91,50,29),(13,64,81,38),(14,37,82,63),(15,62,83,42),(16,41,84,61),(17,66,79,40),(18,39,80,65),(19,20,87,88),(21,24,89,86),(22,85,90,23),(31,32,57,58),(33,36,59,56),(34,55,60,35),(43,44,69,70),(45,48,71,68),(46,67,72,47)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | Q8○D12 |
| kernel | C42.118D6 | C12.6Q8 | C42⋊3S3 | Dic3.D4 | C23.8D6 | C23.9D6 | C23.21D6 | Dic3.Q8 | D6⋊Q8 | C2×Dic3⋊C4 | C12.48D4 | C4×C3⋊D4 | C23.28D6 | C23.23D6 | C23.14D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C42.118D6 ►in GL6(𝔽13)
| 7 | 10 | 0 | 0 | 0 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 12 | 12 | 2 |
| 0 | 0 | 1 | 6 | 11 | 1 |
| 0 | 0 | 1 | 11 | 6 | 1 |
| 0 | 0 | 2 | 12 | 12 | 7 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 7 | 0 |
| 0 | 0 | 0 | 3 | 0 | 7 |
| 0 | 0 | 6 | 0 | 10 | 0 |
| 0 | 0 | 0 | 6 | 0 | 10 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 1 | 6 |
| 0 | 0 | 12 | 12 | 7 | 7 |
| 0 | 0 | 12 | 7 | 2 | 12 |
| 0 | 0 | 6 | 6 | 1 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 6 | 1 |
| 0 | 0 | 12 | 12 | 7 | 7 |
| 0 | 0 | 7 | 12 | 12 | 2 |
| 0 | 0 | 6 | 6 | 1 | 1 |
G:=sub<GL(6,GF(13))| [7,3,0,0,0,0,10,6,0,0,0,0,0,0,7,1,1,2,0,0,12,6,11,12,0,0,12,11,6,12,0,0,2,1,1,7],[4,11,0,0,0,0,2,9,0,0,0,0,0,0,3,0,6,0,0,0,0,3,0,6,0,0,7,0,10,0,0,0,0,7,0,10],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,12,12,6,0,0,1,12,7,6,0,0,1,7,2,1,0,0,6,7,12,1],[12,4,0,0,0,0,0,1,0,0,0,0,0,0,1,12,7,6,0,0,11,12,12,6,0,0,6,7,12,1,0,0,1,7,2,1] >;
C42.118D6 in GAP, Magma, Sage, TeX
C_4^2._{118}D_6 % in TeX
G:=Group("C4^2.118D6"); // GroupNames label
G:=SmallGroup(192,1123);
// by ID
G=gap.SmallGroup(192,1123);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,675,570,297,6278]);
// 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,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations