metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.50D4, M4(2).37D14, D4⋊D7⋊6C4, C8○D4⋊7D7, Q8⋊D7⋊6C4, D4.D7⋊6C4, C7⋊Q16⋊6C4, D4.8(C4×D7), Q8.8(C4×D7), C7⋊4(C8.26D4), C56⋊C4⋊29C2, C14.82(C4×D4), C4○D4.35D14, D28.19(C2×C4), (C2×C8).191D14, C28.448(C2×D4), C8.47(C7⋊D4), D28⋊4C4⋊14C2, D4⋊2Dic7⋊3C2, C28.30(C22×C4), D28.2C4⋊14C2, C28.53D4⋊14C2, (C2×C28).425C23, (C2×C56).237C22, Dic14.19(C2×C4), D4.8D14.3C2, C4○D28.45C22, C22.4(C4○D28), (C4×Dic7).41C22, C4.Dic7.45C22, (C7×M4(2)).40C22, C7⋊C8.6(C2×C4), C4.30(C2×C4×D7), (C7×C8○D4)⋊7C2, C2.27(C4×C7⋊D4), (C7×D4).15(C2×C4), C4.139(C2×C7⋊D4), (C7×Q8).15(C2×C4), (C2×C7⋊C8).144C22, (C2×C14).10(C4○D4), (C7×C4○D4).40C22, (C2×C4).515(C22×D7), SmallGroup(448,679)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.50D4
G = < a,b,c | a56=c2=1, b4=a28, bab-1=a13, cac=a41, cbc=a28b3 >
Subgroups: 412 in 104 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C8⋊C4, C4≀C2, C8.C4, C8○D4, C8○D4, C4○D8, C7⋊C8, C7⋊C8, C56, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8.26D4, C8×D7, C8⋊D7, C2×C7⋊C8, C4.Dic7, C4×Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C4○D28, C7×C4○D4, C56⋊C4, C28.53D4, D28⋊4C4, D4⋊2Dic7, D28.2C4, D4.8D14, C7×C8○D4, C56.50D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C7⋊D4, C22×D7, C8.26D4, C2×C4×D7, C4○D28, C2×C7⋊D4, C4×C7⋊D4, C56.50D4
►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 8 43 50 29 36 15 22)(2 21 44 7 30 49 16 35)(3 34 45 20 31 6 17 48)(4 47 46 33 32 19 18 5)(9 56 51 42 37 28 23 14)(10 13 52 55 38 41 24 27)(11 26 53 12 39 54 25 40)(57 70 71 84 85 98 99 112)(58 83 72 97 86 111 100 69)(59 96 73 110 87 68 101 82)(60 109 74 67 88 81 102 95)(61 66 75 80 89 94 103 108)(62 79 76 93 90 107 104 65)(63 92 77 106 91 64 105 78)
(1 81)(2 66)(3 107)(4 92)(5 77)(6 62)(7 103)(8 88)(9 73)(10 58)(11 99)(12 84)(13 69)(14 110)(15 95)(16 80)(17 65)(18 106)(19 91)(20 76)(21 61)(22 102)(23 87)(24 72)(25 57)(26 98)(27 83)(28 68)(29 109)(30 94)(31 79)(32 64)(33 105)(34 90)(35 75)(36 60)(37 101)(38 86)(39 71)(40 112)(41 97)(42 82)(43 67)(44 108)(45 93)(46 78)(47 63)(48 104)(49 89)(50 74)(51 59)(52 100)(53 85)(54 70)(55 111)(56 96)
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), (1,8,43,50,29,36,15,22)(2,21,44,7,30,49,16,35)(3,34,45,20,31,6,17,48)(4,47,46,33,32,19,18,5)(9,56,51,42,37,28,23,14)(10,13,52,55,38,41,24,27)(11,26,53,12,39,54,25,40)(57,70,71,84,85,98,99,112)(58,83,72,97,86,111,100,69)(59,96,73,110,87,68,101,82)(60,109,74,67,88,81,102,95)(61,66,75,80,89,94,103,108)(62,79,76,93,90,107,104,65)(63,92,77,106,91,64,105,78), (1,81)(2,66)(3,107)(4,92)(5,77)(6,62)(7,103)(8,88)(9,73)(10,58)(11,99)(12,84)(13,69)(14,110)(15,95)(16,80)(17,65)(18,106)(19,91)(20,76)(21,61)(22,102)(23,87)(24,72)(25,57)(26,98)(27,83)(28,68)(29,109)(30,94)(31,79)(32,64)(33,105)(34,90)(35,75)(36,60)(37,101)(38,86)(39,71)(40,112)(41,97)(42,82)(43,67)(44,108)(45,93)(46,78)(47,63)(48,104)(49,89)(50,74)(51,59)(52,100)(53,85)(54,70)(55,111)(56,96)>;
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), (1,8,43,50,29,36,15,22)(2,21,44,7,30,49,16,35)(3,34,45,20,31,6,17,48)(4,47,46,33,32,19,18,5)(9,56,51,42,37,28,23,14)(10,13,52,55,38,41,24,27)(11,26,53,12,39,54,25,40)(57,70,71,84,85,98,99,112)(58,83,72,97,86,111,100,69)(59,96,73,110,87,68,101,82)(60,109,74,67,88,81,102,95)(61,66,75,80,89,94,103,108)(62,79,76,93,90,107,104,65)(63,92,77,106,91,64,105,78), (1,81)(2,66)(3,107)(4,92)(5,77)(6,62)(7,103)(8,88)(9,73)(10,58)(11,99)(12,84)(13,69)(14,110)(15,95)(16,80)(17,65)(18,106)(19,91)(20,76)(21,61)(22,102)(23,87)(24,72)(25,57)(26,98)(27,83)(28,68)(29,109)(30,94)(31,79)(32,64)(33,105)(34,90)(35,75)(36,60)(37,101)(38,86)(39,71)(40,112)(41,97)(42,82)(43,67)(44,108)(45,93)(46,78)(47,63)(48,104)(49,89)(50,74)(51,59)(52,100)(53,85)(54,70)(55,111)(56,96) );
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)], [(1,8,43,50,29,36,15,22),(2,21,44,7,30,49,16,35),(3,34,45,20,31,6,17,48),(4,47,46,33,32,19,18,5),(9,56,51,42,37,28,23,14),(10,13,52,55,38,41,24,27),(11,26,53,12,39,54,25,40),(57,70,71,84,85,98,99,112),(58,83,72,97,86,111,100,69),(59,96,73,110,87,68,101,82),(60,109,74,67,88,81,102,95),(61,66,75,80,89,94,103,108),(62,79,76,93,90,107,104,65),(63,92,77,106,91,64,105,78)], [(1,81),(2,66),(3,107),(4,92),(5,77),(6,62),(7,103),(8,88),(9,73),(10,58),(11,99),(12,84),(13,69),(14,110),(15,95),(16,80),(17,65),(18,106),(19,91),(20,76),(21,61),(22,102),(23,87),(24,72),(25,57),(26,98),(27,83),(28,68),(29,109),(30,94),(31,79),(32,64),(33,105),(34,90),(35,75),(36,60),(37,101),(38,86),(39,71),(40,112),(41,97),(42,82),(43,67),(44,108),(45,93),(46,78),(47,63),(48,104),(49,89),(50,74),(51,59),(52,100),(53,85),(54,70),(55,111),(56,96)]])
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 | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56L | 56M | ··· | 56AD |
| 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 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 1 | 1 | 2 | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | C4×D7 | C4×D7 | C4○D28 | C8.26D4 | C56.50D4 |
| kernel | C56.50D4 | C56⋊C4 | C28.53D4 | D28⋊4C4 | D4⋊2Dic7 | D28.2C4 | D4.8D14 | C7×C8○D4 | D4⋊D7 | D4.D7 | Q8⋊D7 | C7⋊Q16 | C56 | C8○D4 | C2×C14 | C2×C8 | M4(2) | C4○D4 | C8 | D4 | Q8 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 12 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of C56.50D4 ►in GL4(𝔽113) generated by
| 105 | 49 | 0 | 0 |
| 64 | 76 | 0 | 0 |
| 10 | 101 | 37 | 64 |
| 15 | 10 | 49 | 8 |
| 45 | 7 | 0 | 0 |
| 68 | 68 | 0 | 0 |
| 11 | 54 | 64 | 37 |
| 87 | 39 | 8 | 49 |
| 95 | 99 | 0 | 40 |
| 52 | 9 | 40 | 4 |
| 46 | 112 | 80 | 22 |
| 109 | 59 | 14 | 42 |
G:=sub<GL(4,GF(113))| [105,64,10,15,49,76,101,10,0,0,37,49,0,0,64,8],[45,68,11,87,7,68,54,39,0,0,64,8,0,0,37,49],[95,52,46,109,99,9,112,59,0,40,80,14,40,4,22,42] >;
C56.50D4 in GAP, Magma, Sage, TeX
C_{56}._{50}D_4 % in TeX
G:=Group("C56.50D4"); // GroupNames label
G:=SmallGroup(448,679);
// by ID
G=gap.SmallGroup(448,679);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,387,58,136,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=c^2=1,b^4=a^28,b*a*b^-1=a^13,c*a*c=a^41,c*b*c=a^28*b^3>;
// generators/relations