metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56⋊4C4, C56.49D4, C8.27D28, Dic28⋊4C4, C42.23D14, C8.8(C4×D7), C8⋊C4⋊5D7, C56⋊C2⋊2C4, C56.5(C2×C4), C7⋊1(C8.26D4), C4.78(C2×D28), C14.14(C4×D4), C2.17(C4×D28), Dic14⋊C4⋊1C2, D28.15(C2×C4), C28.298(C2×D4), (C2×C8).161D14, C56.C4⋊12C2, D56⋊7C2.8C2, (C4×C28).17C22, D28.2C4⋊12C2, (C2×C56).229C22, C28.108(C22×C4), (C2×C28).792C23, Dic14.15(C2×C4), C4○D28.36C22, C22.21(C4○D28), C4.Dic7.34C22, C4.66(C2×C4×D7), (C7×C8⋊C4)⋊1C2, (C2×C14).63(C4○D4), (C2×C4).682(C22×D7), SmallGroup(448,251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊4C4
G = < a,b,c | a56=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a14b >
Subgroups: 484 in 104 conjugacy classes, 47 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic7, C28, C28, D14, C2×C14, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C7⋊C8, C56, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C8.26D4, C8×D7, C8⋊D7, C56⋊C2, D56, Dic28, C4.Dic7, C4×C28, C2×C56, C4○D28, Dic14⋊C4, C56.C4, C7×C8⋊C4, D28.2C4, D56⋊7C2, D56⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, D28, C22×D7, C8.26D4, C2×C4×D7, C2×D28, C4○D28, C4×D28, D56⋊4C4
►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 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 112)(17 111)(18 110)(19 109)(20 108)(21 107)(22 106)(23 105)(24 104)(25 103)(26 102)(27 101)(28 100)(29 99)(30 98)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 81)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)
(1 15 29 43)(2 44 30 16)(3 17 31 45)(4 46 32 18)(5 19 33 47)(6 48 34 20)(7 21 35 49)(8 50 36 22)(9 23 37 51)(10 52 38 24)(11 25 39 53)(12 54 40 26)(13 27 41 55)(14 56 42 28)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)
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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,99)(30,98)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72), (1,15,29,43)(2,44,30,16)(3,17,31,45)(4,46,32,18)(5,19,33,47)(6,48,34,20)(7,21,35,49)(8,50,36,22)(9,23,37,51)(10,52,38,24)(11,25,39,53)(12,54,40,26)(13,27,41,55)(14,56,42,28)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112)>;
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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,99)(30,98)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,84)(45,83)(46,82)(47,81)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72), (1,15,29,43)(2,44,30,16)(3,17,31,45)(4,46,32,18)(5,19,33,47)(6,48,34,20)(7,21,35,49)(8,50,36,22)(9,23,37,51)(10,52,38,24)(11,25,39,53)(12,54,40,26)(13,27,41,55)(14,56,42,28)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112) );
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)], [(1,71),(2,70),(3,69),(4,68),(5,67),(6,66),(7,65),(8,64),(9,63),(10,62),(11,61),(12,60),(13,59),(14,58),(15,57),(16,112),(17,111),(18,110),(19,109),(20,108),(21,107),(22,106),(23,105),(24,104),(25,103),(26,102),(27,101),(28,100),(29,99),(30,98),(31,97),(32,96),(33,95),(34,94),(35,93),(36,92),(37,91),(38,90),(39,89),(40,88),(41,87),(42,86),(43,85),(44,84),(45,83),(46,82),(47,81),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72)], [(1,15,29,43),(2,44,30,16),(3,17,31,45),(4,46,32,18),(5,19,33,47),(6,48,34,20),(7,21,35,49),(8,50,36,22),(9,23,37,51),(10,52,38,24),(11,25,39,53),(12,54,40,26),(13,27,41,55),(14,56,42,28),(58,86),(60,88),(62,90),(64,92),(66,94),(68,96),(70,98),(72,100),(74,102),(76,104),(78,106),(80,108),(82,110),(84,112)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 28 | 1 | 1 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | D28 | C4○D28 | C8.26D4 | D56⋊4C4 |
| kernel | D56⋊4C4 | Dic14⋊C4 | C56.C4 | C7×C8⋊C4 | D28.2C4 | D56⋊7C2 | C56⋊C2 | D56 | Dic28 | C56 | C8⋊C4 | C2×C14 | C42 | C2×C8 | C8 | C8 | C22 | C7 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 3 | 2 | 3 | 6 | 12 | 12 | 12 | 2 | 12 |
Matrix representation of D56⋊4C4 ►in GL4(𝔽113) generated by
| 102 | 97 | 0 | 0 |
| 43 | 11 | 0 | 0 |
| 108 | 2 | 7 | 112 |
| 14 | 3 | 106 | 106 |
| 51 | 0 | 57 | 8 |
| 78 | 0 | 39 | 51 |
| 108 | 2 | 7 | 112 |
| 92 | 14 | 67 | 55 |
| 98 | 0 | 0 | 0 |
| 63 | 15 | 0 | 0 |
| 30 | 0 | 112 | 0 |
| 86 | 0 | 99 | 1 |
G:=sub<GL(4,GF(113))| [102,43,108,14,97,11,2,3,0,0,7,106,0,0,112,106],[51,78,108,92,0,0,2,14,57,39,7,67,8,51,112,55],[98,63,30,86,0,15,0,0,0,0,112,99,0,0,0,1] >;
D56⋊4C4 in GAP, Magma, Sage, TeX
D_{56}\rtimes_4C_4 % in TeX
G:=Group("D56:4C4"); // GroupNames label
G:=SmallGroup(448,251);
// by ID
G=gap.SmallGroup(448,251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,387,58,136,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^14*b>;
// generators/relations