metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.7D4, D56.4C4, C8.17D28, C28.48D8, C8.2(C4×D7), C56.22(C2×C4), C8.C4⋊1D7, (C2×C28).95D4, (C2×C8).43D14, C28.C8⋊5C2, C4.21(D4⋊D7), C4.4(D14⋊C4), (C2×D56).12C2, C7⋊1(M5(2)⋊C2), (C2×C14).4SD16, C28.4(C22⋊C4), C22.3(Q8⋊D7), C14.7(D4⋊C4), C2.9(C14.D8), (C2×C56).100C22, (C7×C8.C4)⋊9C2, (C2×C4).18(C7⋊D4), SmallGroup(448,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56.C4
G = < a,b,c | a56=b2=1, c4=a28, bab=a-1, cac-1=a43, cbc-1=a7b >
Subgroups: 540 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, D4, C23, D7, C14, C14, C16, C2×C8, M4(2), D8, C2×D4, C28, D14, C2×C14, C8.C4, M5(2), C2×D8, C56, C56, D28, C2×C28, C22×D7, M5(2)⋊C2, C7⋊C16, D56, D56, C2×C56, C7×M4(2), C2×D28, C28.C8, C7×C8.C4, C2×D56, D56.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, D4⋊C4, C4×D7, D28, C7⋊D4, M5(2)⋊C2, D14⋊C4, D4⋊D7, Q8⋊D7, C14.D8, D56.C4
►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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)
(1 71 43 85 29 99 15 57)(2 58 44 72 30 86 16 100)(3 101 45 59 31 73 17 87)(4 88 46 102 32 60 18 74)(5 75 47 89 33 103 19 61)(6 62 48 76 34 90 20 104)(7 105 49 63 35 77 21 91)(8 92 50 106 36 64 22 78)(9 79 51 93 37 107 23 65)(10 66 52 80 38 94 24 108)(11 109 53 67 39 81 25 95)(12 96 54 110 40 68 26 82)(13 83 55 97 41 111 27 69)(14 70 56 84 42 98 28 112)
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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103), (1,71,43,85,29,99,15,57)(2,58,44,72,30,86,16,100)(3,101,45,59,31,73,17,87)(4,88,46,102,32,60,18,74)(5,75,47,89,33,103,19,61)(6,62,48,76,34,90,20,104)(7,105,49,63,35,77,21,91)(8,92,50,106,36,64,22,78)(9,79,51,93,37,107,23,65)(10,66,52,80,38,94,24,108)(11,109,53,67,39,81,25,95)(12,96,54,110,40,68,26,82)(13,83,55,97,41,111,27,69)(14,70,56,84,42,98,28,112)>;
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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103), (1,71,43,85,29,99,15,57)(2,58,44,72,30,86,16,100)(3,101,45,59,31,73,17,87)(4,88,46,102,32,60,18,74)(5,75,47,89,33,103,19,61)(6,62,48,76,34,90,20,104)(7,105,49,63,35,77,21,91)(8,92,50,106,36,64,22,78)(9,79,51,93,37,107,23,65)(10,66,52,80,38,94,24,108)(11,109,53,67,39,81,25,95)(12,96,54,110,40,68,26,82)(13,83,55,97,41,111,27,69)(14,70,56,84,42,98,28,112) );
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)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(57,91),(58,90),(59,89),(60,88),(61,87),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103)], [(1,71,43,85,29,99,15,57),(2,58,44,72,30,86,16,100),(3,101,45,59,31,73,17,87),(4,88,46,102,32,60,18,74),(5,75,47,89,33,103,19,61),(6,62,48,76,34,90,20,104),(7,105,49,63,35,77,21,91),(8,92,50,106,36,64,22,78),(9,79,51,93,37,107,23,65),(10,66,52,80,38,94,24,108),(11,109,53,67,39,81,25,95),(12,96,54,110,40,68,26,82),(13,83,55,97,41,111,27,69),(14,70,56,84,42,98,28,112)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4×D7 | D28 | C7⋊D4 | M5(2)⋊C2 | D4⋊D7 | Q8⋊D7 | D56.C4 |
| kernel | D56.C4 | C28.C8 | C7×C8.C4 | C2×D56 | D56 | C56 | C2×C28 | C8.C4 | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D56.C4 ►in GL4(𝔽113) generated by
| 52 | 44 | 0 | 0 |
| 69 | 91 | 0 | 0 |
| 0 | 0 | 33 | 106 |
| 0 | 0 | 7 | 91 |
| 80 | 7 | 0 | 0 |
| 6 | 33 | 0 | 0 |
| 0 | 0 | 36 | 17 |
| 0 | 0 | 90 | 77 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 96 | 105 | 0 | 0 |
| 8 | 17 | 0 | 0 |
G:=sub<GL(4,GF(113))| [52,69,0,0,44,91,0,0,0,0,33,7,0,0,106,91],[80,6,0,0,7,33,0,0,0,0,36,90,0,0,17,77],[0,0,96,8,0,0,105,17,1,0,0,0,0,1,0,0] >;
D56.C4 in GAP, Magma, Sage, TeX
D_{56}.C_4 % in TeX
G:=Group("D56.C4"); // GroupNames label
G:=SmallGroup(448,52);
// by ID
G=gap.SmallGroup(448,52);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,184,675,794,192,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=1,c^4=a^28,b*a*b=a^-1,c*a*c^-1=a^43,c*b*c^-1=a^7*b>;
// generators/relations