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, C8⋊D14⋊4C2, D56⋊C2⋊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, C56⋊C2⋊7C22, C8⋊D7⋊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⋊2D7⋊4C22, (Q8×C14)⋊22C22, (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
Generators and relations for C56.C23
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 >
Subgroups: 1388 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, D28, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, C22×D7, D4○SD16, C8×D7, C8⋊D7, C56⋊C2, D56, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, Q8×D7, Q8×D7, Q8⋊2D7, Q8⋊2D7, Q8×C14, C7×C4○D4, D28.C4, C8⋊D14, D7×SD16, D56⋊C2, Q16⋊D7, Q8.D14, C2×Q8⋊D7, D4.8D14, C7×C8.C22, Q8.10D14, D4⋊8D14, C56.C23
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○SD16, D4×D7, C23×D7, C2×D4×D7, C56.C23
►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)
(1 102)(2 59)(3 72)(4 85)(5 98)(6 111)(7 68)(8 81)(9 94)(10 107)(11 64)(12 77)(13 90)(14 103)(15 60)(16 73)(17 86)(18 99)(19 112)(20 69)(21 82)(22 95)(23 108)(24 65)(25 78)(26 91)(27 104)(28 61)(29 74)(30 87)(31 100)(32 57)(33 70)(34 83)(35 96)(36 109)(37 66)(38 79)(39 92)(40 105)(41 62)(42 75)(43 88)(44 101)(45 58)(46 71)(47 84)(48 97)(49 110)(50 67)(51 80)(52 93)(53 106)(54 63)(55 76)(56 89)
(1 67 29 95)(2 82 30 110)(3 97 31 69)(4 112 32 84)(5 71 33 99)(6 86 34 58)(7 101 35 73)(8 60 36 88)(9 75 37 103)(10 90 38 62)(11 105 39 77)(12 64 40 92)(13 79 41 107)(14 94 42 66)(15 109 43 81)(16 68 44 96)(17 83 45 111)(18 98 46 70)(19 57 47 85)(20 72 48 100)(21 87 49 59)(22 102 50 74)(23 61 51 89)(24 76 52 104)(25 91 53 63)(26 106 54 78)(27 65 55 93)(28 80 56 108)
(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)(57 106 85 78)(58 93 86 65)(59 80 87 108)(60 67 88 95)(61 110 89 82)(62 97 90 69)(63 84 91 112)(64 71 92 99)(66 101 94 73)(68 75 96 103)(70 105 98 77)(72 79 100 107)(74 109 102 81)(76 83 104 111)
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), (1,102)(2,59)(3,72)(4,85)(5,98)(6,111)(7,68)(8,81)(9,94)(10,107)(11,64)(12,77)(13,90)(14,103)(15,60)(16,73)(17,86)(18,99)(19,112)(20,69)(21,82)(22,95)(23,108)(24,65)(25,78)(26,91)(27,104)(28,61)(29,74)(30,87)(31,100)(32,57)(33,70)(34,83)(35,96)(36,109)(37,66)(38,79)(39,92)(40,105)(41,62)(42,75)(43,88)(44,101)(45,58)(46,71)(47,84)(48,97)(49,110)(50,67)(51,80)(52,93)(53,106)(54,63)(55,76)(56,89), (1,67,29,95)(2,82,30,110)(3,97,31,69)(4,112,32,84)(5,71,33,99)(6,86,34,58)(7,101,35,73)(8,60,36,88)(9,75,37,103)(10,90,38,62)(11,105,39,77)(12,64,40,92)(13,79,41,107)(14,94,42,66)(15,109,43,81)(16,68,44,96)(17,83,45,111)(18,98,46,70)(19,57,47,85)(20,72,48,100)(21,87,49,59)(22,102,50,74)(23,61,51,89)(24,76,52,104)(25,91,53,63)(26,106,54,78)(27,65,55,93)(28,80,56,108), (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)(57,106,85,78)(58,93,86,65)(59,80,87,108)(60,67,88,95)(61,110,89,82)(62,97,90,69)(63,84,91,112)(64,71,92,99)(66,101,94,73)(68,75,96,103)(70,105,98,77)(72,79,100,107)(74,109,102,81)(76,83,104,111)>;
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), (1,102)(2,59)(3,72)(4,85)(5,98)(6,111)(7,68)(8,81)(9,94)(10,107)(11,64)(12,77)(13,90)(14,103)(15,60)(16,73)(17,86)(18,99)(19,112)(20,69)(21,82)(22,95)(23,108)(24,65)(25,78)(26,91)(27,104)(28,61)(29,74)(30,87)(31,100)(32,57)(33,70)(34,83)(35,96)(36,109)(37,66)(38,79)(39,92)(40,105)(41,62)(42,75)(43,88)(44,101)(45,58)(46,71)(47,84)(48,97)(49,110)(50,67)(51,80)(52,93)(53,106)(54,63)(55,76)(56,89), (1,67,29,95)(2,82,30,110)(3,97,31,69)(4,112,32,84)(5,71,33,99)(6,86,34,58)(7,101,35,73)(8,60,36,88)(9,75,37,103)(10,90,38,62)(11,105,39,77)(12,64,40,92)(13,79,41,107)(14,94,42,66)(15,109,43,81)(16,68,44,96)(17,83,45,111)(18,98,46,70)(19,57,47,85)(20,72,48,100)(21,87,49,59)(22,102,50,74)(23,61,51,89)(24,76,52,104)(25,91,53,63)(26,106,54,78)(27,65,55,93)(28,80,56,108), (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)(57,106,85,78)(58,93,86,65)(59,80,87,108)(60,67,88,95)(61,110,89,82)(62,97,90,69)(63,84,91,112)(64,71,92,99)(66,101,94,73)(68,75,96,103)(70,105,98,77)(72,79,100,107)(74,109,102,81)(76,83,104,111) );
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)], [(1,102),(2,59),(3,72),(4,85),(5,98),(6,111),(7,68),(8,81),(9,94),(10,107),(11,64),(12,77),(13,90),(14,103),(15,60),(16,73),(17,86),(18,99),(19,112),(20,69),(21,82),(22,95),(23,108),(24,65),(25,78),(26,91),(27,104),(28,61),(29,74),(30,87),(31,100),(32,57),(33,70),(34,83),(35,96),(36,109),(37,66),(38,79),(39,92),(40,105),(41,62),(42,75),(43,88),(44,101),(45,58),(46,71),(47,84),(48,97),(49,110),(50,67),(51,80),(52,93),(53,106),(54,63),(55,76),(56,89)], [(1,67,29,95),(2,82,30,110),(3,97,31,69),(4,112,32,84),(5,71,33,99),(6,86,34,58),(7,101,35,73),(8,60,36,88),(9,75,37,103),(10,90,38,62),(11,105,39,77),(12,64,40,92),(13,79,41,107),(14,94,42,66),(15,109,43,81),(16,68,44,96),(17,83,45,111),(18,98,46,70),(19,57,47,85),(20,72,48,100),(21,87,49,59),(22,102,50,74),(23,61,51,89),(24,76,52,104),(25,91,53,63),(26,106,54,78),(27,65,55,93),(28,80,56,108)], [(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),(57,106,85,78),(58,93,86,65),(59,80,87,108),(60,67,88,95),(61,110,89,82),(62,97,90,69),(63,84,91,112),(64,71,92,99),(66,101,94,73),(68,75,96,103),(70,105,98,77),(72,79,100,107),(74,109,102,81),(76,83,104,111)]])
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 |
Matrix representation of C56.C23 ►in 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] >;
C56.C23 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