metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56:7C4, Dic28:7C4, M4(2).27D14, C7:3(C8oD8), C7:C8.38D4, C56:C2:6C4, C8.16(C4xD7), C56.35(C2xC4), (C8xDic7):1C2, C4.213(D4xD7), C14.56(C4xD4), C8.C4:8D7, D28:4C4:8C2, D28.C4:11C2, D28.11(C2xC4), (C2xC8).252D14, C28.372(C2xD4), D56:7C2.5C2, (C2xC56).42C22, C28.55(C22xC4), (C2xC28).311C23, Dic14.11(C2xC4), C4oD28.18C22, C2.16(D28:C4), C22.2(Q8:2D7), (C4xDic7).235C22, (C7xM4(2)).21C22, C4.47(C2xC4xD7), (C7xC8.C4):5C2, (C2xC14).2(C4oD4), (C2xC7:C8).238C22, (C2xC4).414(C22xD7), SmallGroup(448,429)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56:7C4
G = < a,b,c | a56=b2=c4=1, bab=a-1, cac-1=a41, cbc-1=a54b >
Subgroups: 508 in 106 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2xC4, C2xC4, D4, Q8, D7, C14, C14, C42, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C4oD4, Dic7, C28, D14, C2xC14, C4xC8, C4wrC2, C8.C4, C8oD4, C4oD8, C7:C8, C56, C56, Dic14, C4xD7, D28, C2xDic7, C7:D4, C2xC28, C8oD8, C8xD7, C8:D7, C56:C2, D56, Dic28, C2xC7:C8, C4xDic7, C2xC56, C7xM4(2), C4oD28, C8xDic7, D28:4C4, C7xC8.C4, D56:7C2, D28.C4, D56:7C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D7, C22xC4, C2xD4, C4oD4, D14, C4xD4, C4xD7, C22xD7, C8oD8, C2xC4xD7, D4xD7, Q8:2D7, D28:C4, D56:7C4
►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 61)(2 60)(3 59)(4 58)(5 57)(6 112)(7 111)(8 110)(9 109)(10 108)(11 107)(12 106)(13 105)(14 104)(15 103)(16 102)(17 101)(18 100)(19 99)(20 98)(21 97)(22 96)(23 95)(24 94)(25 93)(26 92)(27 91)(28 90)(29 89)(30 88)(31 87)(32 86)(33 85)(34 84)(35 83)(36 82)(37 81)(38 80)(39 79)(40 78)(41 77)(42 76)(43 75)(44 74)(45 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)
(1 29)(2 14)(3 55)(4 40)(5 25)(6 10)(7 51)(8 36)(9 21)(11 47)(12 32)(13 17)(15 43)(16 28)(18 54)(19 39)(20 24)(22 50)(23 35)(26 46)(27 31)(30 42)(33 53)(34 38)(37 49)(41 45)(44 56)(48 52)(57 67 85 95)(58 108 86 80)(59 93 87 65)(60 78 88 106)(61 63 89 91)(62 104 90 76)(64 74 92 102)(66 100 94 72)(68 70 96 98)(69 111 97 83)(71 81 99 109)(73 107 101 79)(75 77 103 105)(82 84 110 112)
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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,112)(7,111)(8,110)(9,109)(10,108)(11,107)(12,106)(13,105)(14,104)(15,103)(16,102)(17,101)(18,100)(19,99)(20,98)(21,97)(22,96)(23,95)(24,94)(25,93)(26,92)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,84)(35,83)(36,82)(37,81)(38,80)(39,79)(40,78)(41,77)(42,76)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62), (1,29)(2,14)(3,55)(4,40)(5,25)(6,10)(7,51)(8,36)(9,21)(11,47)(12,32)(13,17)(15,43)(16,28)(18,54)(19,39)(20,24)(22,50)(23,35)(26,46)(27,31)(30,42)(33,53)(34,38)(37,49)(41,45)(44,56)(48,52)(57,67,85,95)(58,108,86,80)(59,93,87,65)(60,78,88,106)(61,63,89,91)(62,104,90,76)(64,74,92,102)(66,100,94,72)(68,70,96,98)(69,111,97,83)(71,81,99,109)(73,107,101,79)(75,77,103,105)(82,84,110,112)>;
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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,112)(7,111)(8,110)(9,109)(10,108)(11,107)(12,106)(13,105)(14,104)(15,103)(16,102)(17,101)(18,100)(19,99)(20,98)(21,97)(22,96)(23,95)(24,94)(25,93)(26,92)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,84)(35,83)(36,82)(37,81)(38,80)(39,79)(40,78)(41,77)(42,76)(43,75)(44,74)(45,73)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62), (1,29)(2,14)(3,55)(4,40)(5,25)(6,10)(7,51)(8,36)(9,21)(11,47)(12,32)(13,17)(15,43)(16,28)(18,54)(19,39)(20,24)(22,50)(23,35)(26,46)(27,31)(30,42)(33,53)(34,38)(37,49)(41,45)(44,56)(48,52)(57,67,85,95)(58,108,86,80)(59,93,87,65)(60,78,88,106)(61,63,89,91)(62,104,90,76)(64,74,92,102)(66,100,94,72)(68,70,96,98)(69,111,97,83)(71,81,99,109)(73,107,101,79)(75,77,103,105)(82,84,110,112) );
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)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,112),(7,111),(8,110),(9,109),(10,108),(11,107),(12,106),(13,105),(14,104),(15,103),(16,102),(17,101),(18,100),(19,99),(20,98),(21,97),(22,96),(23,95),(24,94),(25,93),(26,92),(27,91),(28,90),(29,89),(30,88),(31,87),(32,86),(33,85),(34,84),(35,83),(36,82),(37,81),(38,80),(39,79),(40,78),(41,77),(42,76),(43,75),(44,74),(45,73),(46,72),(47,71),(48,70),(49,69),(50,68),(51,67),(52,66),(53,65),(54,64),(55,63),(56,62)], [(1,29),(2,14),(3,55),(4,40),(5,25),(6,10),(7,51),(8,36),(9,21),(11,47),(12,32),(13,17),(15,43),(16,28),(18,54),(19,39),(20,24),(22,50),(23,35),(26,46),(27,31),(30,42),(33,53),(34,38),(37,49),(41,45),(44,56),(48,52),(57,67,85,95),(58,108,86,80),(59,93,87,65),(60,78,88,106),(61,63,89,91),(62,104,90,76),(64,74,92,102),(66,100,94,72),(68,70,96,98),(69,111,97,83),(71,81,99,109),(73,107,101,79),(75,77,103,105),(82,84,110,112)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 28 | 1 | 1 | 2 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | C4oD4 | D14 | D14 | C4xD7 | C8oD8 | D4xD7 | Q8:2D7 | D56:7C4 |
| kernel | D56:7C4 | C8xDic7 | D28:4C4 | C7xC8.C4 | D56:7C2 | D28.C4 | C56:C2 | D56 | Dic28 | C7:C8 | C8.C4 | C2xC14 | C2xC8 | M4(2) | C8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 3 | 2 | 3 | 6 | 12 | 8 | 3 | 3 | 12 |
Matrix representation of D56:7C4 ►in GL4(F113) generated by
| 79 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 44 |
| 79 | 112 | 0 | 0 |
| 25 | 34 | 0 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 15 | 0 |
| 1 | 0 | 0 | 0 |
| 79 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(113))| [79,1,0,0,112,0,0,0,0,0,18,0,0,0,0,44],[79,25,0,0,112,34,0,0,0,0,0,15,0,0,98,0],[1,79,0,0,0,112,0,0,0,0,112,0,0,0,0,15] >;
D56:7C4 in GAP, Magma, Sage, TeX
D_{56}\rtimes_7C_4 % in TeX
G:=Group("D56:7C4"); // GroupNames label
G:=SmallGroup(448,429);
// by ID
G=gap.SmallGroup(448,429);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,555,58,136,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^41,c*b*c^-1=a^54*b>;
// generators/relations