metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16⋊5D14, SD16⋊7D14, D28.43D4, D56⋊4C22, C56.6C23, C28.25C24, M4(2)⋊13D14, Dic14.43D4, D28.18C23, Dic14.18C23, C7⋊D4.6D4, D56⋊C2⋊4C2, C8⋊D14⋊4C2, D28.C4⋊3C2, (C2×Q8)⋊12D14, (D7×SD16)⋊4C2, C7⋊5(D4○SD16), C4.117(D4×D7), (C8×D7)⋊6C22, C7⋊C8.27C23, C8.C22⋊4D7, Q8.D14⋊2C2, D4⋊8D14⋊8C2, (Q8×D7)⋊4C22, C8.6(C22×D7), D4⋊D7⋊16C22, Q16⋊D7⋊3C2, C4○D4.14D14, D14.34(C2×D4), C28.246(C2×D4), C4○D28⋊9C22, C8⋊D7⋊7C22, C56⋊C2⋊7C22, Q8⋊D7⋊15C22, (C7×Q16)⋊3C22, (D4×D7).4C22, C22.16(D4×D7), C4.25(C23×D7), D4.8D14⋊5C2, D4.D7⋊15C22, Dic7.39(C2×D4), (Q8×C14)⋊22C22, Q8⋊2D7⋊4C22, (C7×SD16)⋊7C22, (C4×D7).16C23, D4.18(C22×D7), C7⋊Q16⋊14C22, (C7×D4).18C23, Q8.18(C22×D7), (C7×Q8).18C23, (C2×C28).116C23, Q8.10D14⋊5C2, C14.126(C22×D4), (C7×M4(2))⋊7C22, (C2×D28).182C22, C2.99(C2×D4×D7), (C2×C7⋊C8)⋊19C22, (C2×Q8⋊D7)⋊29C2, (C2×C14).71(C2×D4), (C7×C8.C22)⋊3C2, (C7×C4○D4).27C22, (C2×C4).100(C22×D7), SmallGroup(448,1231)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1388 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×15], Q8, Q8 [×2], Q8 [×5], C23 [×3], D7 [×5], C14, C14 [×2], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×3], SD16 [×2], SD16 [×8], Q16 [×2], Q16, C2×D4 [×6], C2×Q8, C2×Q8 [×3], C4○D4, C4○D4 [×10], Dic7 [×2], Dic7, C28 [×2], C28 [×3], D14 [×2], D14 [×6], C2×C14, C2×C14, C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22, C8.C22 [×2], 2+ (1+4), 2- (1+4), C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14 [×2], C4×D7 [×2], C4×D7 [×7], D28 [×2], D28 [×2], D28 [×5], C7⋊D4 [×2], C7⋊D4 [×3], C2×C28, C2×C28 [×2], C7×D4, C7×D4, C7×Q8, C7×Q8 [×2], C7×Q8, C22×D7 [×3], D4○SD16, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], D56 [×2], C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, Q8⋊D7 [×4], C7⋊Q16, C7×M4(2), C7×SD16 [×2], C7×Q16 [×2], C2×D28, C2×D28, C4○D28 [×2], C4○D28 [×3], D4×D7 [×2], D4×D7 [×2], Q8×D7 [×2], Q8×D7, Q8⋊2D7 [×4], Q8⋊2D7, Q8×C14, C7×C4○D4, D28.C4, C8⋊D14, D7×SD16 [×2], D56⋊C2 [×2], Q16⋊D7 [×2], Q8.D14 [×2], C2×Q8⋊D7, D4.8D14, C7×C8.C22, Q8.10D14, D4⋊8D14, C56.C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D4○SD16, D4×D7 [×2], C23×D7, C2×D4×D7, C56.C23
Generators and relations
G = < a,b,c,d | a56=b2=1, c2=d2=a28, bab=a13, cac-1=a15, dad-1=a43, bc=cb, dbd-1=a28b, 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)
(1 73)(2 86)(3 99)(4 112)(5 69)(6 82)(7 95)(8 108)(9 65)(10 78)(11 91)(12 104)(13 61)(14 74)(15 87)(16 100)(17 57)(18 70)(19 83)(20 96)(21 109)(22 66)(23 79)(24 92)(25 105)(26 62)(27 75)(28 88)(29 101)(30 58)(31 71)(32 84)(33 97)(34 110)(35 67)(36 80)(37 93)(38 106)(39 63)(40 76)(41 89)(42 102)(43 59)(44 72)(45 85)(46 98)(47 111)(48 68)(49 81)(50 94)(51 107)(52 64)(53 77)(54 90)(55 103)(56 60)
(1 94 29 66)(2 109 30 81)(3 68 31 96)(4 83 32 111)(5 98 33 70)(6 57 34 85)(7 72 35 100)(8 87 36 59)(9 102 37 74)(10 61 38 89)(11 76 39 104)(12 91 40 63)(13 106 41 78)(14 65 42 93)(15 80 43 108)(16 95 44 67)(17 110 45 82)(18 69 46 97)(19 84 47 112)(20 99 48 71)(21 58 49 86)(22 73 50 101)(23 88 51 60)(24 103 52 75)(25 62 53 90)(26 77 54 105)(27 92 55 64)(28 107 56 79)
(1 36 29 8)(2 23 30 51)(3 10 31 38)(4 53 32 25)(5 40 33 12)(6 27 34 55)(7 14 35 42)(9 44 37 16)(11 18 39 46)(13 48 41 20)(15 22 43 50)(17 52 45 24)(19 26 47 54)(21 56 49 28)(57 92 85 64)(58 79 86 107)(59 66 87 94)(60 109 88 81)(61 96 89 68)(62 83 90 111)(63 70 91 98)(65 100 93 72)(67 74 95 102)(69 104 97 76)(71 78 99 106)(73 108 101 80)(75 82 103 110)(77 112 105 84)
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), (1,73)(2,86)(3,99)(4,112)(5,69)(6,82)(7,95)(8,108)(9,65)(10,78)(11,91)(12,104)(13,61)(14,74)(15,87)(16,100)(17,57)(18,70)(19,83)(20,96)(21,109)(22,66)(23,79)(24,92)(25,105)(26,62)(27,75)(28,88)(29,101)(30,58)(31,71)(32,84)(33,97)(34,110)(35,67)(36,80)(37,93)(38,106)(39,63)(40,76)(41,89)(42,102)(43,59)(44,72)(45,85)(46,98)(47,111)(48,68)(49,81)(50,94)(51,107)(52,64)(53,77)(54,90)(55,103)(56,60), (1,94,29,66)(2,109,30,81)(3,68,31,96)(4,83,32,111)(5,98,33,70)(6,57,34,85)(7,72,35,100)(8,87,36,59)(9,102,37,74)(10,61,38,89)(11,76,39,104)(12,91,40,63)(13,106,41,78)(14,65,42,93)(15,80,43,108)(16,95,44,67)(17,110,45,82)(18,69,46,97)(19,84,47,112)(20,99,48,71)(21,58,49,86)(22,73,50,101)(23,88,51,60)(24,103,52,75)(25,62,53,90)(26,77,54,105)(27,92,55,64)(28,107,56,79), (1,36,29,8)(2,23,30,51)(3,10,31,38)(4,53,32,25)(5,40,33,12)(6,27,34,55)(7,14,35,42)(9,44,37,16)(11,18,39,46)(13,48,41,20)(15,22,43,50)(17,52,45,24)(19,26,47,54)(21,56,49,28)(57,92,85,64)(58,79,86,107)(59,66,87,94)(60,109,88,81)(61,96,89,68)(62,83,90,111)(63,70,91,98)(65,100,93,72)(67,74,95,102)(69,104,97,76)(71,78,99,106)(73,108,101,80)(75,82,103,110)(77,112,105,84)>;
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), (1,73)(2,86)(3,99)(4,112)(5,69)(6,82)(7,95)(8,108)(9,65)(10,78)(11,91)(12,104)(13,61)(14,74)(15,87)(16,100)(17,57)(18,70)(19,83)(20,96)(21,109)(22,66)(23,79)(24,92)(25,105)(26,62)(27,75)(28,88)(29,101)(30,58)(31,71)(32,84)(33,97)(34,110)(35,67)(36,80)(37,93)(38,106)(39,63)(40,76)(41,89)(42,102)(43,59)(44,72)(45,85)(46,98)(47,111)(48,68)(49,81)(50,94)(51,107)(52,64)(53,77)(54,90)(55,103)(56,60), (1,94,29,66)(2,109,30,81)(3,68,31,96)(4,83,32,111)(5,98,33,70)(6,57,34,85)(7,72,35,100)(8,87,36,59)(9,102,37,74)(10,61,38,89)(11,76,39,104)(12,91,40,63)(13,106,41,78)(14,65,42,93)(15,80,43,108)(16,95,44,67)(17,110,45,82)(18,69,46,97)(19,84,47,112)(20,99,48,71)(21,58,49,86)(22,73,50,101)(23,88,51,60)(24,103,52,75)(25,62,53,90)(26,77,54,105)(27,92,55,64)(28,107,56,79), (1,36,29,8)(2,23,30,51)(3,10,31,38)(4,53,32,25)(5,40,33,12)(6,27,34,55)(7,14,35,42)(9,44,37,16)(11,18,39,46)(13,48,41,20)(15,22,43,50)(17,52,45,24)(19,26,47,54)(21,56,49,28)(57,92,85,64)(58,79,86,107)(59,66,87,94)(60,109,88,81)(61,96,89,68)(62,83,90,111)(63,70,91,98)(65,100,93,72)(67,74,95,102)(69,104,97,76)(71,78,99,106)(73,108,101,80)(75,82,103,110)(77,112,105,84) );
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)], [(1,73),(2,86),(3,99),(4,112),(5,69),(6,82),(7,95),(8,108),(9,65),(10,78),(11,91),(12,104),(13,61),(14,74),(15,87),(16,100),(17,57),(18,70),(19,83),(20,96),(21,109),(22,66),(23,79),(24,92),(25,105),(26,62),(27,75),(28,88),(29,101),(30,58),(31,71),(32,84),(33,97),(34,110),(35,67),(36,80),(37,93),(38,106),(39,63),(40,76),(41,89),(42,102),(43,59),(44,72),(45,85),(46,98),(47,111),(48,68),(49,81),(50,94),(51,107),(52,64),(53,77),(54,90),(55,103),(56,60)], [(1,94,29,66),(2,109,30,81),(3,68,31,96),(4,83,32,111),(5,98,33,70),(6,57,34,85),(7,72,35,100),(8,87,36,59),(9,102,37,74),(10,61,38,89),(11,76,39,104),(12,91,40,63),(13,106,41,78),(14,65,42,93),(15,80,43,108),(16,95,44,67),(17,110,45,82),(18,69,46,97),(19,84,47,112),(20,99,48,71),(21,58,49,86),(22,73,50,101),(23,88,51,60),(24,103,52,75),(25,62,53,90),(26,77,54,105),(27,92,55,64),(28,107,56,79)], [(1,36,29,8),(2,23,30,51),(3,10,31,38),(4,53,32,25),(5,40,33,12),(6,27,34,55),(7,14,35,42),(9,44,37,16),(11,18,39,46),(13,48,41,20),(15,22,43,50),(17,52,45,24),(19,26,47,54),(21,56,49,28),(57,92,85,64),(58,79,86,107),(59,66,87,94),(60,109,88,81),(61,96,89,68),(62,83,90,111),(63,70,91,98),(65,100,93,72),(67,74,95,102),(69,104,97,76),(71,78,99,106),(73,108,101,80),(75,82,103,110),(77,112,105,84)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 80 | 112 | 0 | 0 | 0 | 0 |
| 2 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 100 | 0 | 0 |
| 0 | 0 | 13 | 100 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 8 | 80 | 0 | 0 | 0 | 0 |
| 43 | 105 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 0 | 100 | 100 |
| 0 | 0 | 100 | 13 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 100 | 100 | 0 | 0 |
| 0 | 0 | 100 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 100 |
| 0 | 0 | 0 | 0 | 100 | 13 |
G:=sub<GL(6,GF(113))| [80,2,0,0,0,0,112,24,0,0,0,0,0,0,100,13,0,0,0,0,100,100,0,0,0,0,0,0,13,13,0,0,0,0,100,13],[8,43,0,0,0,0,80,105,0,0,0,0,0,0,0,0,100,13,0,0,0,0,13,13,0,0,13,100,0,0,0,0,100,100,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,100,100,0,0,0,0,100,13,0,0,0,0,0,0,100,100,0,0,0,0,100,13] >;
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D4○SD16 | D4×D7 | D4×D7 | C56.C23 |
| kernel | C56.C23 | D28.C4 | C8⋊D14 | D7×SD16 | D56⋊C2 | Q16⋊D7 | Q8.D14 | C2×Q8⋊D7 | D4.8D14 | C7×C8.C22 | Q8.10D14 | D4⋊8D14 | Dic14 | D28 | C7⋊D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
In GAP, Magma, Sage, TeX
C_{56}.C_2^3 % in TeX
G:=Group("C56.C2^3"); // GroupNames label
G:=SmallGroup(448,1231);
// by ID
G=gap.SmallGroup(448,1231);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,184,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^56=b^2=1,c^2=d^2=a^28,b*a*b=a^13,c*a*c^-1=a^15,d*a*d^-1=a^43,b*c=c*b,d*b*d^-1=a^28*b,c*d=d*c>;
// generators/relations