metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.42D4, C8.24D28, D28.22D4, Dic14.22D4, M4(2).11D14, C4.58(C2×D28), (C2×C8).72D14, C8.C4⋊7D7, C4.137(D4×D7), C8⋊D14.2C2, C28.138(C2×D4), C7⋊3(D4.3D4), D28.2C4⋊9C2, C8.D14⋊10C2, C28.46D4⋊4C2, C28.47D4⋊4C2, C14.51(C4⋊D4), C2.24(C28⋊1D4), (C2×C56).155C22, (C2×C28).314C23, C4○D28.41C22, (C2×D28).89C22, C22.8(Q8⋊2D7), (C7×M4(2)).8C22, C4.Dic7.39C22, (C2×Dic14).95C22, (C7×C8.C4)⋊8C2, (C2×C56⋊C2)⋊26C2, (C2×C14).5(C4○D4), (C2×C4).115(C22×D7), SmallGroup(448,432)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.42D4
G = < a,b,c | a56=1, b4=c2=a28, bab-1=a43, cac-1=a27, cbc-1=b3 >
Subgroups: 668 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C7, C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, D7 [×2], C14, C14, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, Dic7 [×2], C28 [×2], D14 [×3], C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C56 [×2], C56 [×2], Dic14, Dic14 [×2], C4×D7, D28, D28 [×2], C2×Dic7, C7⋊D4, C2×C28, C22×D7, D4.3D4, C8×D7, C8⋊D7, C56⋊C2 [×4], D56, Dic28, C4.Dic7, C2×C56, C7×M4(2) [×2], C2×Dic14, C2×D28, C4○D28, C28.46D4, C28.47D4, C7×C8.C4, D28.2C4, C2×C56⋊C2, C8⋊D14, C8.D14, C56.42D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D7, C2×D4 [×2], C4○D4, D14 [×3], C4⋊D4, D28 [×2], C22×D7, D4.3D4, C2×D28, D4×D7, Q8⋊2D7, C28⋊1D4, C56.42D4
►On 112 points
(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 86 15 72 29 58 43 100)(2 73 16 59 30 101 44 87)(3 60 17 102 31 88 45 74)(4 103 18 89 32 75 46 61)(5 90 19 76 33 62 47 104)(6 77 20 63 34 105 48 91)(7 64 21 106 35 92 49 78)(8 107 22 93 36 79 50 65)(9 94 23 80 37 66 51 108)(10 81 24 67 38 109 52 95)(11 68 25 110 39 96 53 82)(12 111 26 97 40 83 54 69)(13 98 27 84 41 70 55 112)(14 85 28 71 42 57 56 99)
(1 22 29 50)(2 49 30 21)(3 20 31 48)(4 47 32 19)(5 18 33 46)(6 45 34 17)(7 16 35 44)(8 43 36 15)(9 14 37 42)(10 41 38 13)(11 12 39 40)(23 56 51 28)(24 27 52 55)(25 54 53 26)(57 80 85 108)(58 107 86 79)(59 78 87 106)(60 105 88 77)(61 76 89 104)(62 103 90 75)(63 74 91 102)(64 101 92 73)(65 72 93 100)(66 99 94 71)(67 70 95 98)(68 97 96 69)(81 112 109 84)(82 83 110 111)
G:=sub<Sym(112)| (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,86,15,72,29,58,43,100)(2,73,16,59,30,101,44,87)(3,60,17,102,31,88,45,74)(4,103,18,89,32,75,46,61)(5,90,19,76,33,62,47,104)(6,77,20,63,34,105,48,91)(7,64,21,106,35,92,49,78)(8,107,22,93,36,79,50,65)(9,94,23,80,37,66,51,108)(10,81,24,67,38,109,52,95)(11,68,25,110,39,96,53,82)(12,111,26,97,40,83,54,69)(13,98,27,84,41,70,55,112)(14,85,28,71,42,57,56,99), (1,22,29,50)(2,49,30,21)(3,20,31,48)(4,47,32,19)(5,18,33,46)(6,45,34,17)(7,16,35,44)(8,43,36,15)(9,14,37,42)(10,41,38,13)(11,12,39,40)(23,56,51,28)(24,27,52,55)(25,54,53,26)(57,80,85,108)(58,107,86,79)(59,78,87,106)(60,105,88,77)(61,76,89,104)(62,103,90,75)(63,74,91,102)(64,101,92,73)(65,72,93,100)(66,99,94,71)(67,70,95,98)(68,97,96,69)(81,112,109,84)(82,83,110,111)>;
G:=Group( (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,86,15,72,29,58,43,100)(2,73,16,59,30,101,44,87)(3,60,17,102,31,88,45,74)(4,103,18,89,32,75,46,61)(5,90,19,76,33,62,47,104)(6,77,20,63,34,105,48,91)(7,64,21,106,35,92,49,78)(8,107,22,93,36,79,50,65)(9,94,23,80,37,66,51,108)(10,81,24,67,38,109,52,95)(11,68,25,110,39,96,53,82)(12,111,26,97,40,83,54,69)(13,98,27,84,41,70,55,112)(14,85,28,71,42,57,56,99), (1,22,29,50)(2,49,30,21)(3,20,31,48)(4,47,32,19)(5,18,33,46)(6,45,34,17)(7,16,35,44)(8,43,36,15)(9,14,37,42)(10,41,38,13)(11,12,39,40)(23,56,51,28)(24,27,52,55)(25,54,53,26)(57,80,85,108)(58,107,86,79)(59,78,87,106)(60,105,88,77)(61,76,89,104)(62,103,90,75)(63,74,91,102)(64,101,92,73)(65,72,93,100)(66,99,94,71)(67,70,95,98)(68,97,96,69)(81,112,109,84)(82,83,110,111) );
G=PermutationGroup([(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),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,86,15,72,29,58,43,100),(2,73,16,59,30,101,44,87),(3,60,17,102,31,88,45,74),(4,103,18,89,32,75,46,61),(5,90,19,76,33,62,47,104),(6,77,20,63,34,105,48,91),(7,64,21,106,35,92,49,78),(8,107,22,93,36,79,50,65),(9,94,23,80,37,66,51,108),(10,81,24,67,38,109,52,95),(11,68,25,110,39,96,53,82),(12,111,26,97,40,83,54,69),(13,98,27,84,41,70,55,112),(14,85,28,71,42,57,56,99)], [(1,22,29,50),(2,49,30,21),(3,20,31,48),(4,47,32,19),(5,18,33,46),(6,45,34,17),(7,16,35,44),(8,43,36,15),(9,14,37,42),(10,41,38,13),(11,12,39,40),(23,56,51,28),(24,27,52,55),(25,54,53,26),(57,80,85,108),(58,107,86,79),(59,78,87,106),(60,105,88,77),(61,76,89,104),(62,103,90,75),(63,74,91,102),(64,101,92,73),(65,72,93,100),(66,99,94,71),(67,70,95,98),(68,97,96,69),(81,112,109,84),(82,83,110,111)])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 56 | 2 | 2 | 28 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 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 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D28 | D4.3D4 | D4×D7 | Q8⋊2D7 | C56.42D4 |
| kernel | C56.42D4 | C28.46D4 | C28.47D4 | C7×C8.C4 | D28.2C4 | C2×C56⋊C2 | C8⋊D14 | C8.D14 | C56 | Dic14 | D28 | C8.C4 | C2×C14 | C2×C8 | M4(2) | C8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56.42D4 ►in GL4(𝔽113) generated by
| 17 | 105 | 0 | 0 |
| 95 | 35 | 0 | 0 |
| 18 | 9 | 104 | 95 |
| 111 | 104 | 18 | 11 |
| 0 | 0 | 112 | 1 |
| 89 | 1 | 111 | 24 |
| 82 | 81 | 112 | 0 |
| 73 | 73 | 112 | 0 |
| 35 | 8 | 0 | 0 |
| 101 | 78 | 0 | 0 |
| 112 | 104 | 18 | 9 |
| 19 | 9 | 102 | 95 |
G:=sub<GL(4,GF(113))| [17,95,18,111,105,35,9,104,0,0,104,18,0,0,95,11],[0,89,82,73,0,1,81,73,112,111,112,112,1,24,0,0],[35,101,112,19,8,78,104,9,0,0,18,102,0,0,9,95] >;
C56.42D4 in GAP, Magma, Sage, TeX
C_{56}._{42}D_4 % in TeX
G:=Group("C56.42D4"); // GroupNames label
G:=SmallGroup(448,432);
// by ID
G=gap.SmallGroup(448,432);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,555,58,1123,136,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=1,b^4=c^2=a^28,b*a*b^-1=a^43,c*a*c^-1=a^27,c*b*c^-1=b^3>;
// generators/relations