metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.48D4, C28.25M4(2), C23.(C7⋊C8), (C2×C56).2C4, (C2×C28).1C8, C7⋊2(C23.C8), (C2×C8).2Dic7, (C2×C8).153D14, C8.33(C7⋊D4), (C22×C28).7C4, (C22×C14).2C8, C28.C8⋊11C2, (C2×M4(2)).4D7, C4.7(C4.Dic7), (C22×C4).4Dic7, C14.18(C22⋊C8), C28.93(C22⋊C4), (C2×C56).222C22, (C14×M4(2)).4C2, C4.26(C23.D7), C2.7(C28.55D4), (C2×C4).(C7⋊C8), C22.4(C2×C7⋊C8), (C2×C14).32(C2×C8), (C2×C28).303(C2×C4), (C2×C4).75(C2×Dic7), SmallGroup(448,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.D4
G = < a,b,c | a56=1, b4=a42, c2=a49, bab-1=a13, cac-1=a41, cbc-1=a7b3 >
Subgroups: 132 in 58 conjugacy classes, 31 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C16, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, M5(2), C2×M4(2), C56, C56, C2×C28, C2×C28, C22×C14, C23.C8, C7⋊C16, C2×C56, C7×M4(2), C22×C28, C28.C8, C14×M4(2), C56.D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), Dic7, D14, C22⋊C8, C7⋊C8, C2×Dic7, C7⋊D4, C23.C8, C2×C7⋊C8, C4.Dic7, C23.D7, C28.55D4, C56.D4
►On 112 points
Generators in S112
(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 99 50 64 43 85 36 106 29 71 22 92 15 57 8 78)(2 112 51 77 44 98 37 63 30 84 23 105 16 70 9 91)(3 69 52 90 45 111 38 76 31 97 24 62 17 83 10 104)(4 82 53 103 46 68 39 89 32 110 25 75 18 96 11 61)(5 95 54 60 47 81 40 102 33 67 26 88 19 109 12 74)(6 108 55 73 48 94 41 59 34 80 27 101 20 66 13 87)(7 65 56 86 49 107 42 72 35 93 28 58 21 79 14 100)
(1 71 50 64 43 57 36 106 29 99 22 92 15 85 8 78)(2 112 51 105 44 98 37 91 30 84 23 77 16 70 9 63)(3 97 52 90 45 83 38 76 31 69 24 62 17 111 10 104)(4 82 53 75 46 68 39 61 32 110 25 103 18 96 11 89)(5 67 54 60 47 109 40 102 33 95 26 88 19 81 12 74)(6 108 55 101 48 94 41 87 34 80 27 73 20 66 13 59)(7 93 56 86 49 79 42 72 35 65 28 58 21 107 14 100)
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), (1,99,50,64,43,85,36,106,29,71,22,92,15,57,8,78)(2,112,51,77,44,98,37,63,30,84,23,105,16,70,9,91)(3,69,52,90,45,111,38,76,31,97,24,62,17,83,10,104)(4,82,53,103,46,68,39,89,32,110,25,75,18,96,11,61)(5,95,54,60,47,81,40,102,33,67,26,88,19,109,12,74)(6,108,55,73,48,94,41,59,34,80,27,101,20,66,13,87)(7,65,56,86,49,107,42,72,35,93,28,58,21,79,14,100), (1,71,50,64,43,57,36,106,29,99,22,92,15,85,8,78)(2,112,51,105,44,98,37,91,30,84,23,77,16,70,9,63)(3,97,52,90,45,83,38,76,31,69,24,62,17,111,10,104)(4,82,53,75,46,68,39,61,32,110,25,103,18,96,11,89)(5,67,54,60,47,109,40,102,33,95,26,88,19,81,12,74)(6,108,55,101,48,94,41,87,34,80,27,73,20,66,13,59)(7,93,56,86,49,79,42,72,35,65,28,58,21,107,14,100)>;
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), (1,99,50,64,43,85,36,106,29,71,22,92,15,57,8,78)(2,112,51,77,44,98,37,63,30,84,23,105,16,70,9,91)(3,69,52,90,45,111,38,76,31,97,24,62,17,83,10,104)(4,82,53,103,46,68,39,89,32,110,25,75,18,96,11,61)(5,95,54,60,47,81,40,102,33,67,26,88,19,109,12,74)(6,108,55,73,48,94,41,59,34,80,27,101,20,66,13,87)(7,65,56,86,49,107,42,72,35,93,28,58,21,79,14,100), (1,71,50,64,43,57,36,106,29,99,22,92,15,85,8,78)(2,112,51,105,44,98,37,91,30,84,23,77,16,70,9,63)(3,97,52,90,45,83,38,76,31,69,24,62,17,111,10,104)(4,82,53,75,46,68,39,61,32,110,25,103,18,96,11,89)(5,67,54,60,47,109,40,102,33,95,26,88,19,81,12,74)(6,108,55,101,48,94,41,87,34,80,27,73,20,66,13,59)(7,93,56,86,49,79,42,72,35,65,28,58,21,107,14,100) );
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)], [(1,99,50,64,43,85,36,106,29,71,22,92,15,57,8,78),(2,112,51,77,44,98,37,63,30,84,23,105,16,70,9,91),(3,69,52,90,45,111,38,76,31,97,24,62,17,83,10,104),(4,82,53,103,46,68,39,89,32,110,25,75,18,96,11,61),(5,95,54,60,47,81,40,102,33,67,26,88,19,109,12,74),(6,108,55,73,48,94,41,59,34,80,27,101,20,66,13,87),(7,65,56,86,49,107,42,72,35,93,28,58,21,79,14,100)], [(1,71,50,64,43,57,36,106,29,99,22,92,15,85,8,78),(2,112,51,105,44,98,37,91,30,84,23,77,16,70,9,63),(3,97,52,90,45,83,38,76,31,69,24,62,17,111,10,104),(4,82,53,75,46,68,39,61,32,110,25,103,18,96,11,89),(5,67,54,60,47,109,40,102,33,95,26,88,19,81,12,74),(6,108,55,101,48,94,41,87,34,80,27,73,20,66,13,59),(7,93,56,86,49,79,42,72,35,65,28,58,21,107,14,100)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | ··· | 14I | 14J | ··· | 14O | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D7 | M4(2) | Dic7 | D14 | Dic7 | C7⋊D4 | C7⋊C8 | C7⋊C8 | C4.Dic7 | C23.C8 | C56.D4 |
| kernel | C56.D4 | C28.C8 | C14×M4(2) | C2×C56 | C22×C28 | C2×C28 | C22×C14 | C56 | C2×M4(2) | C28 | C2×C8 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C7 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 3 | 2 | 3 | 3 | 3 | 12 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of C56.D4 ►in GL4(𝔽113) generated by
| 0 | 30 | 0 | 0 |
| 111 | 0 | 0 | 0 |
| 0 | 0 | 0 | 49 |
| 0 | 0 | 57 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 1 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 15 | 0 | 0 | 0 |
G:=sub<GL(4,GF(113))| [0,111,0,0,30,0,0,0,0,0,0,57,0,0,49,0],[0,0,0,98,0,0,1,0,1,0,0,0,0,112,0,0],[0,0,0,15,0,0,1,0,1,0,0,0,0,1,0,0] >;
C56.D4 in GAP, Magma, Sage, TeX
C_{56}.D_4 % in TeX
G:=Group("C56.D4"); // GroupNames label
G:=SmallGroup(448,110);
// by ID
G=gap.SmallGroup(448,110);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,387,100,1123,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=1,b^4=a^42,c^2=a^49,b*a*b^-1=a^13,c*a*c^-1=a^41,c*b*c^-1=a^7*b^3>;
// generators/relations