metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.9C23, D56⋊10C22, C28.60C24, C23.21D28, M4(2)⋊19D14, Dic28⋊9C22, D28.23C23, Dic14.23C23, (C2×C8)⋊5D14, (C2×C56)⋊8C22, C4.73(C2×D28), C8⋊D14⋊13C2, C8.9(C22×D7), C28.239(C2×D4), (C2×C4).157D28, (C2×C28).205D4, (C2×M4(2))⋊5D7, D56⋊7C2⋊10C2, C4.57(C23×D7), C8.D14⋊13C2, C4○D28⋊17C22, (C2×D28)⋊53C22, C56⋊C2⋊10C22, C7⋊1(D8⋊C22), (C14×M4(2))⋊5C2, C22.22(C2×D28), C2.29(C22×D28), C14.27(C22×D4), (C2×C28).798C23, (C22×C4).267D14, (C22×C14).120D4, (C2×Dic14)⋊64C22, (C7×M4(2))⋊21C22, (C22×C28).268C22, (C2×C4○D28)⋊27C2, (C2×C14).64(C2×D4), (C2×C4).225(C22×D7), SmallGroup(448,1201)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1380 in 262 conjugacy classes, 107 normal (21 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], D7 [×4], C14, C14 [×3], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], Dic7 [×4], C28 [×2], C28 [×2], D14 [×8], C2×C14, C2×C14 [×2], C2×C14, C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C56 [×4], Dic14 [×4], Dic14 [×2], C4×D7 [×8], D28 [×4], D28 [×2], C2×Dic7 [×2], C7⋊D4 [×8], C2×C28 [×2], C2×C28 [×4], C22×D7 [×2], C22×C14, D8⋊C22, C56⋊C2 [×8], D56 [×4], Dic28 [×4], C2×C56 [×2], C7×M4(2) [×4], C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28 [×2], C4○D28 [×8], C4○D28 [×4], C2×C7⋊D4 [×2], C22×C28, D56⋊7C2 [×4], C8⋊D14 [×4], C8.D14 [×4], C14×M4(2), C2×C4○D28 [×2], C56.9C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, D28 [×4], C22×D7 [×7], D8⋊C22, C2×D28 [×6], C23×D7, C22×D28, C56.9C23
Generators and relations
G = < a,b,c,d | a56=b2=1, c2=d2=a28, bab=a27, ac=ca, dad-1=a29, bc=cb, bd=db, cd=dc >
►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)
(2 28)(3 55)(4 26)(5 53)(6 24)(7 51)(8 22)(9 49)(10 20)(11 47)(12 18)(13 45)(14 16)(15 43)(17 41)(19 39)(21 37)(23 35)(25 33)(27 31)(30 56)(32 54)(34 52)(36 50)(38 48)(40 46)(42 44)(57 69)(58 96)(59 67)(60 94)(61 65)(62 92)(64 90)(66 88)(68 86)(70 84)(71 111)(72 82)(73 109)(74 80)(75 107)(76 78)(77 105)(79 103)(81 101)(83 99)(85 97)(87 95)(89 93)(98 112)(100 110)(102 108)(104 106)
(1 63 29 91)(2 64 30 92)(3 65 31 93)(4 66 32 94)(5 67 33 95)(6 68 34 96)(7 69 35 97)(8 70 36 98)(9 71 37 99)(10 72 38 100)(11 73 39 101)(12 74 40 102)(13 75 41 103)(14 76 42 104)(15 77 43 105)(16 78 44 106)(17 79 45 107)(18 80 46 108)(19 81 47 109)(20 82 48 110)(21 83 49 111)(22 84 50 112)(23 85 51 57)(24 86 52 58)(25 87 53 59)(26 88 54 60)(27 89 55 61)(28 90 56 62)
(1 63 29 91)(2 92 30 64)(3 65 31 93)(4 94 32 66)(5 67 33 95)(6 96 34 68)(7 69 35 97)(8 98 36 70)(9 71 37 99)(10 100 38 72)(11 73 39 101)(12 102 40 74)(13 75 41 103)(14 104 42 76)(15 77 43 105)(16 106 44 78)(17 79 45 107)(18 108 46 80)(19 81 47 109)(20 110 48 82)(21 83 49 111)(22 112 50 84)(23 85 51 57)(24 58 52 86)(25 87 53 59)(26 60 54 88)(27 89 55 61)(28 62 56 90)
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), (2,28)(3,55)(4,26)(5,53)(6,24)(7,51)(8,22)(9,49)(10,20)(11,47)(12,18)(13,45)(14,16)(15,43)(17,41)(19,39)(21,37)(23,35)(25,33)(27,31)(30,56)(32,54)(34,52)(36,50)(38,48)(40,46)(42,44)(57,69)(58,96)(59,67)(60,94)(61,65)(62,92)(64,90)(66,88)(68,86)(70,84)(71,111)(72,82)(73,109)(74,80)(75,107)(76,78)(77,105)(79,103)(81,101)(83,99)(85,97)(87,95)(89,93)(98,112)(100,110)(102,108)(104,106), (1,63,29,91)(2,64,30,92)(3,65,31,93)(4,66,32,94)(5,67,33,95)(6,68,34,96)(7,69,35,97)(8,70,36,98)(9,71,37,99)(10,72,38,100)(11,73,39,101)(12,74,40,102)(13,75,41,103)(14,76,42,104)(15,77,43,105)(16,78,44,106)(17,79,45,107)(18,80,46,108)(19,81,47,109)(20,82,48,110)(21,83,49,111)(22,84,50,112)(23,85,51,57)(24,86,52,58)(25,87,53,59)(26,88,54,60)(27,89,55,61)(28,90,56,62), (1,63,29,91)(2,92,30,64)(3,65,31,93)(4,94,32,66)(5,67,33,95)(6,96,34,68)(7,69,35,97)(8,98,36,70)(9,71,37,99)(10,100,38,72)(11,73,39,101)(12,102,40,74)(13,75,41,103)(14,104,42,76)(15,77,43,105)(16,106,44,78)(17,79,45,107)(18,108,46,80)(19,81,47,109)(20,110,48,82)(21,83,49,111)(22,112,50,84)(23,85,51,57)(24,58,52,86)(25,87,53,59)(26,60,54,88)(27,89,55,61)(28,62,56,90)>;
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), (2,28)(3,55)(4,26)(5,53)(6,24)(7,51)(8,22)(9,49)(10,20)(11,47)(12,18)(13,45)(14,16)(15,43)(17,41)(19,39)(21,37)(23,35)(25,33)(27,31)(30,56)(32,54)(34,52)(36,50)(38,48)(40,46)(42,44)(57,69)(58,96)(59,67)(60,94)(61,65)(62,92)(64,90)(66,88)(68,86)(70,84)(71,111)(72,82)(73,109)(74,80)(75,107)(76,78)(77,105)(79,103)(81,101)(83,99)(85,97)(87,95)(89,93)(98,112)(100,110)(102,108)(104,106), (1,63,29,91)(2,64,30,92)(3,65,31,93)(4,66,32,94)(5,67,33,95)(6,68,34,96)(7,69,35,97)(8,70,36,98)(9,71,37,99)(10,72,38,100)(11,73,39,101)(12,74,40,102)(13,75,41,103)(14,76,42,104)(15,77,43,105)(16,78,44,106)(17,79,45,107)(18,80,46,108)(19,81,47,109)(20,82,48,110)(21,83,49,111)(22,84,50,112)(23,85,51,57)(24,86,52,58)(25,87,53,59)(26,88,54,60)(27,89,55,61)(28,90,56,62), (1,63,29,91)(2,92,30,64)(3,65,31,93)(4,94,32,66)(5,67,33,95)(6,96,34,68)(7,69,35,97)(8,98,36,70)(9,71,37,99)(10,100,38,72)(11,73,39,101)(12,102,40,74)(13,75,41,103)(14,104,42,76)(15,77,43,105)(16,106,44,78)(17,79,45,107)(18,108,46,80)(19,81,47,109)(20,110,48,82)(21,83,49,111)(22,112,50,84)(23,85,51,57)(24,58,52,86)(25,87,53,59)(26,60,54,88)(27,89,55,61)(28,62,56,90) );
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)], [(2,28),(3,55),(4,26),(5,53),(6,24),(7,51),(8,22),(9,49),(10,20),(11,47),(12,18),(13,45),(14,16),(15,43),(17,41),(19,39),(21,37),(23,35),(25,33),(27,31),(30,56),(32,54),(34,52),(36,50),(38,48),(40,46),(42,44),(57,69),(58,96),(59,67),(60,94),(61,65),(62,92),(64,90),(66,88),(68,86),(70,84),(71,111),(72,82),(73,109),(74,80),(75,107),(76,78),(77,105),(79,103),(81,101),(83,99),(85,97),(87,95),(89,93),(98,112),(100,110),(102,108),(104,106)], [(1,63,29,91),(2,64,30,92),(3,65,31,93),(4,66,32,94),(5,67,33,95),(6,68,34,96),(7,69,35,97),(8,70,36,98),(9,71,37,99),(10,72,38,100),(11,73,39,101),(12,74,40,102),(13,75,41,103),(14,76,42,104),(15,77,43,105),(16,78,44,106),(17,79,45,107),(18,80,46,108),(19,81,47,109),(20,82,48,110),(21,83,49,111),(22,84,50,112),(23,85,51,57),(24,86,52,58),(25,87,53,59),(26,88,54,60),(27,89,55,61),(28,90,56,62)], [(1,63,29,91),(2,92,30,64),(3,65,31,93),(4,94,32,66),(5,67,33,95),(6,96,34,68),(7,69,35,97),(8,98,36,70),(9,71,37,99),(10,100,38,72),(11,73,39,101),(12,102,40,74),(13,75,41,103),(14,104,42,76),(15,77,43,105),(16,106,44,78),(17,79,45,107),(18,108,46,80),(19,81,47,109),(20,110,48,82),(21,83,49,111),(22,112,50,84),(23,85,51,57),(24,58,52,86),(25,87,53,59),(26,60,54,88),(27,89,55,61),(28,62,56,90)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 15 | 0 | 63 |
| 0 | 0 | 98 | 104 | 0 | 0 |
| 0 | 0 | 15 | 9 | 0 | 98 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 89 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 112 | 0 | 0 |
| 0 | 0 | 0 | 91 | 112 | 111 |
| 0 | 0 | 112 | 22 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 9 | 0 | 98 |
G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,24,0,0,0,0,0,0,0,0,98,15,0,0,0,15,104,9,0,0,98,0,0,0,0,0,0,63,0,98],[112,89,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,112,0,0,0,112,91,22,0,0,0,0,112,0,0,0,0,0,111,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,9,0,0,0,0,98,0,0,0,0,0,0,98] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | D8⋊C22 | C56.9C23 |
| kernel | C56.9C23 | D56⋊7C2 | C8⋊D14 | C8.D14 | C14×M4(2) | C2×C4○D28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 4 | 4 | 4 | 1 | 2 | 3 | 1 | 3 | 6 | 12 | 3 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{56}._9C_2^3 % in TeX
G:=Group("C56.9C2^3"); // GroupNames label
G:=SmallGroup(448,1201);
// by ID
G=gap.SmallGroup(448,1201);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,570,80,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^56=b^2=1,c^2=d^2=a^28,b*a*b=a^27,a*c=c*a,d*a*d^-1=a^29,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations