metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56⋊2C4, C56.86D4, Dic28⋊2C4, M5(2)⋊6D7, C28.7SD16, C22.3D56, C8.6(C4×D7), C56.3(C2×C4), C8⋊Dic7⋊1C2, (C2×C14).2D8, C7⋊2(D8⋊2C4), (C2×C4).11D28, (C2×C8).48D14, (C2×C28).101D4, C8.43(C7⋊D4), C4.20(D14⋊C4), D56⋊7C2.7C2, C4.12(C56⋊C2), (C7×M5(2))⋊10C2, (C2×C56).52C22, C28.44(C22⋊C4), C2.11(C2.D56), C14.19(D4⋊C4), SmallGroup(448,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊2C4
G = < a,b,c | a56=b2=c4=1, bab=a-1, cac-1=a27, cbc-1=a47b >
Subgroups: 436 in 58 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic7, C28, D14, C2×C14, C4.Q8, M5(2), C4○D8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, D8⋊2C4, C112, C56⋊C2, D56, Dic28, C4⋊Dic7, C2×C56, C4○D28, C8⋊Dic7, C7×M5(2), D56⋊7C2, D56⋊2C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, D14, D4⋊C4, C4×D7, D28, C7⋊D4, D8⋊2C4, C56⋊C2, D56, D14⋊C4, C2.D56, D56⋊2C4
►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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 112)(9 111)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)
(1 15)(2 42)(3 13)(4 40)(5 11)(6 38)(7 9)(8 36)(10 34)(12 32)(14 30)(16 28)(17 55)(18 26)(19 53)(20 24)(21 51)(23 49)(25 47)(27 45)(29 43)(31 41)(33 39)(35 37)(44 56)(46 54)(48 52)(57 92 85 64)(58 63 86 91)(59 90 87 62)(60 61 88 89)(65 84 93 112)(66 111 94 83)(67 82 95 110)(68 109 96 81)(69 80 97 108)(70 107 98 79)(71 78 99 106)(72 105 100 77)(73 76 101 104)(74 103 102 75)
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64), (1,15)(2,42)(3,13)(4,40)(5,11)(6,38)(7,9)(8,36)(10,34)(12,32)(14,30)(16,28)(17,55)(18,26)(19,53)(20,24)(21,51)(23,49)(25,47)(27,45)(29,43)(31,41)(33,39)(35,37)(44,56)(46,54)(48,52)(57,92,85,64)(58,63,86,91)(59,90,87,62)(60,61,88,89)(65,84,93,112)(66,111,94,83)(67,82,95,110)(68,109,96,81)(69,80,97,108)(70,107,98,79)(71,78,99,106)(72,105,100,77)(73,76,101,104)(74,103,102,75)>;
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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,112)(9,111)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64), (1,15)(2,42)(3,13)(4,40)(5,11)(6,38)(7,9)(8,36)(10,34)(12,32)(14,30)(16,28)(17,55)(18,26)(19,53)(20,24)(21,51)(23,49)(25,47)(27,45)(29,43)(31,41)(33,39)(35,37)(44,56)(46,54)(48,52)(57,92,85,64)(58,63,86,91)(59,90,87,62)(60,61,88,89)(65,84,93,112)(66,111,94,83)(67,82,95,110)(68,109,96,81)(69,80,97,108)(70,107,98,79)(71,78,99,106)(72,105,100,77)(73,76,101,104)(74,103,102,75) );
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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,112),(9,111),(10,110),(11,109),(12,108),(13,107),(14,106),(15,105),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,77),(44,76),(45,75),(46,74),(47,73),(48,72),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64)], [(1,15),(2,42),(3,13),(4,40),(5,11),(6,38),(7,9),(8,36),(10,34),(12,32),(14,30),(16,28),(17,55),(18,26),(19,53),(20,24),(21,51),(23,49),(25,47),(27,45),(29,43),(31,41),(33,39),(35,37),(44,56),(46,54),(48,52),(57,92,85,64),(58,63,86,91),(59,90,87,62),(60,61,88,89),(65,84,93,112),(66,111,94,83),(67,82,95,110),(68,109,96,81),(69,80,97,108),(70,107,98,79),(71,78,99,106),(72,105,100,77),(73,76,101,104),(74,103,102,75)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 56 | 2 | 2 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | SD16 | D8 | D14 | C4×D7 | C7⋊D4 | D28 | C56⋊C2 | D56 | D8⋊2C4 | D56⋊2C4 |
| kernel | D56⋊2C4 | C8⋊Dic7 | C7×M5(2) | D56⋊7C2 | D56 | Dic28 | C56 | C2×C28 | M5(2) | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of D56⋊2C4 ►in GL6(𝔽113)
| 79 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 97 | 0 | 0 |
| 0 | 0 | 7 | 87 | 0 | 0 |
| 0 | 0 | 0 | 59 | 13 | 13 |
| 0 | 0 | 3 | 54 | 100 | 13 |
| 79 | 112 | 0 | 0 | 0 | 0 |
| 25 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 16 | 97 |
| 0 | 0 | 110 | 0 | 0 | 87 |
| 0 | 0 | 24 | 13 | 54 | 59 |
| 0 | 0 | 82 | 13 | 59 | 54 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 55 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 77 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 77 | 48 | 100 | 13 |
| 0 | 0 | 39 | 65 | 13 | 13 |
G:=sub<GL(6,GF(113))| [79,1,0,0,0,0,112,0,0,0,0,0,0,0,0,7,0,3,0,0,97,87,59,54,0,0,0,0,13,100,0,0,0,0,13,13],[79,25,0,0,0,0,112,34,0,0,0,0,0,0,5,110,24,82,0,0,0,0,13,13,0,0,16,0,54,59,0,0,97,87,59,54],[15,55,0,0,0,0,0,98,0,0,0,0,0,0,1,0,77,39,0,0,77,112,48,65,0,0,0,0,100,13,0,0,0,0,13,13] >;
D56⋊2C4 in GAP, Magma, Sage, TeX
D_{56}\rtimes_2C_4 % in TeX
G:=Group("D56:2C4"); // GroupNames label
G:=SmallGroup(448,75);
// by ID
G=gap.SmallGroup(448,75);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,422,387,268,570,136,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^27,c*b*c^-1=a^47*b>;
// generators/relations