metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.51D6, D4.S3⋊5C4, D4.6(C4×S3), (C4×D4).8S3, C6.72(C4×D4), C4⋊C4.246D6, (D4×C12).9C2, Dic6⋊11(C2×C4), (C4×Dic6)⋊20C2, (C2×D4).193D6, C6.Q16⋊32C2, (C2×C12).256D4, C4.39(C4○D12), C12.53(C4○D4), C6.SD16⋊30C2, C3⋊5(SD16⋊C4), C6.88(C8⋊C22), C42.S3⋊6C2, (C4×C12).89C22, C12.24(C22×C4), (C2×C12).340C23, C2.4(D12⋊6C22), D4⋊Dic3.10C2, C2.3(Q8.14D6), (C6×D4).235C22, C6.108(C8.C22), C4⋊Dic3.329C22, (C2×Dic6).265C22, C3⋊C8⋊9(C2×C4), C4.24(S3×C2×C4), C2.18(C4×C3⋊D4), (C3×D4).13(C2×C4), (C2×C6).471(C2×D4), (C2×C3⋊C8).95C22, (C2×D4.S3).4C2, C22.78(C2×C3⋊D4), (C2×C4).219(C3⋊D4), (C3×C4⋊C4).277C22, (C2×C4).440(C22×S3), SmallGroup(192,577)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.51D6
G = < a,b,c,d | a4=b4=c6=1, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 280 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, SD16⋊C4, C42.S3, C6.Q16, C6.SD16, D4⋊Dic3, C4×Dic6, C2×D4.S3, D4×C12, C42.51D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C8⋊C22, C8.C22, S3×C2×C4, C4○D12, C2×C3⋊D4, SD16⋊C4, C4×C3⋊D4, D12⋊6C22, Q8.14D6, C42.51D6
►On 96 points
(1 86 24 62)(2 83 17 59)(3 88 18 64)(4 85 19 61)(5 82 20 58)(6 87 21 63)(7 84 22 60)(8 81 23 57)(9 42 79 71)(10 47 80 68)(11 44 73 65)(12 41 74 70)(13 46 75 67)(14 43 76 72)(15 48 77 69)(16 45 78 66)(25 50 90 34)(26 55 91 39)(27 52 92 36)(28 49 93 33)(29 54 94 38)(30 51 95 35)(31 56 96 40)(32 53 89 37)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)
(1 26 65)(2 68 27 4 66 29)(3 32 67 7 28 71)(5 30 69)(6 72 31 8 70 25)(9 84 53)(10 56 85 12 54 87)(11 82 55 15 86 51)(13 88 49)(14 52 81 16 50 83)(17 47 92 19 45 94)(18 89 46 22 93 42)(20 95 48)(21 43 96 23 41 90)(24 91 44)(33 75 64)(34 59 76 36 57 78)(35 73 58 39 77 62)(37 79 60)(38 63 80 40 61 74)
(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)
G:=sub<Sym(96)| (1,86,24,62)(2,83,17,59)(3,88,18,64)(4,85,19,61)(5,82,20,58)(6,87,21,63)(7,84,22,60)(8,81,23,57)(9,42,79,71)(10,47,80,68)(11,44,73,65)(12,41,74,70)(13,46,75,67)(14,43,76,72)(15,48,77,69)(16,45,78,66)(25,50,90,34)(26,55,91,39)(27,52,92,36)(28,49,93,33)(29,54,94,38)(30,51,95,35)(31,56,96,40)(32,53,89,37), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96), (1,26,65)(2,68,27,4,66,29)(3,32,67,7,28,71)(5,30,69)(6,72,31,8,70,25)(9,84,53)(10,56,85,12,54,87)(11,82,55,15,86,51)(13,88,49)(14,52,81,16,50,83)(17,47,92,19,45,94)(18,89,46,22,93,42)(20,95,48)(21,43,96,23,41,90)(24,91,44)(33,75,64)(34,59,76,36,57,78)(35,73,58,39,77,62)(37,79,60)(38,63,80,40,61,74), (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)>;
G:=Group( (1,86,24,62)(2,83,17,59)(3,88,18,64)(4,85,19,61)(5,82,20,58)(6,87,21,63)(7,84,22,60)(8,81,23,57)(9,42,79,71)(10,47,80,68)(11,44,73,65)(12,41,74,70)(13,46,75,67)(14,43,76,72)(15,48,77,69)(16,45,78,66)(25,50,90,34)(26,55,91,39)(27,52,92,36)(28,49,93,33)(29,54,94,38)(30,51,95,35)(31,56,96,40)(32,53,89,37), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96), (1,26,65)(2,68,27,4,66,29)(3,32,67,7,28,71)(5,30,69)(6,72,31,8,70,25)(9,84,53)(10,56,85,12,54,87)(11,82,55,15,86,51)(13,88,49)(14,52,81,16,50,83)(17,47,92,19,45,94)(18,89,46,22,93,42)(20,95,48)(21,43,96,23,41,90)(24,91,44)(33,75,64)(34,59,76,36,57,78)(35,73,58,39,77,62)(37,79,60)(38,63,80,40,61,74), (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) );
G=PermutationGroup([[(1,86,24,62),(2,83,17,59),(3,88,18,64),(4,85,19,61),(5,82,20,58),(6,87,21,63),(7,84,22,60),(8,81,23,57),(9,42,79,71),(10,47,80,68),(11,44,73,65),(12,41,74,70),(13,46,75,67),(14,43,76,72),(15,48,77,69),(16,45,78,66),(25,50,90,34),(26,55,91,39),(27,52,92,36),(28,49,93,33),(29,54,94,38),(30,51,95,35),(31,56,96,40),(32,53,89,37)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96)], [(1,26,65),(2,68,27,4,66,29),(3,32,67,7,28,71),(5,30,69),(6,72,31,8,70,25),(9,84,53),(10,56,85,12,54,87),(11,82,55,15,86,51),(13,88,49),(14,52,81,16,50,83),(17,47,92,19,45,94),(18,89,46,22,93,42),(20,95,48),(21,43,96,23,41,90),(24,91,44),(33,75,64),(34,59,76,36,57,78),(35,73,58,39,77,62),(37,79,60),(38,63,80,40,61,74)], [(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)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | ··· | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4○D12 | C8⋊C22 | C8.C22 | D12⋊6C22 | Q8.14D6 |
| kernel | C42.51D6 | C42.S3 | C6.Q16 | C6.SD16 | D4⋊Dic3 | C4×Dic6 | C2×D4.S3 | D4×C12 | D4.S3 | C4×D4 | C2×C12 | C42 | C4⋊C4 | C2×D4 | C12 | C2×C4 | D4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C42.51D6 ►in GL6(𝔽73)
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 14 |
| 0 | 0 | 0 | 0 | 59 | 7 |
| 0 | 0 | 7 | 59 | 0 | 0 |
| 0 | 0 | 14 | 66 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 46 | 59 | 27 |
| 0 | 0 | 60 | 59 | 13 | 14 |
| 0 | 0 | 14 | 46 | 14 | 46 |
| 0 | 0 | 60 | 59 | 60 | 59 |
G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,66,59,0,0,0,0,14,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[9,0,0,0,0,0,0,65,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,64,0,0,0,0,8,0,0,0,0,0,0,0,14,60,14,60,0,0,46,59,46,59,0,0,59,13,14,60,0,0,27,14,46,59] >;
C42.51D6 in GAP, Magma, Sage, TeX
C_4^2._{51}D_6 % in TeX
G:=Group("C4^2.51D6"); // GroupNames label
G:=SmallGroup(192,577);
// by ID
G=gap.SmallGroup(192,577);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,387,58,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations