metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.23D4, D28.25D4, Dic14.25D4, (C14×D8)⋊2C2, (C2×D8)⋊10D7, C4.61(D4×D7), (C2×C8).87D14, C56.C4⋊3C2, (C2×D4).67D14, C28.D4⋊7C2, C28.169(C2×D4), C7⋊4(D4.4D4), C8.27(C7⋊D4), D28.2C4⋊1C2, D4.D14⋊4C2, (C2×C56).32C22, C2.18(C28⋊2D4), (C2×C28).437C23, C4○D28.46C22, (D4×C14).86C22, C14.111(C4⋊D4), C4.Dic7.19C22, C22.20(D4⋊2D7), C4.81(C2×C7⋊D4), (C2×C4).126(C22×D7), (C2×C14).158(C4○D4), SmallGroup(448,694)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.23D4
G = < a,b,c | a56=c2=1, b4=a28, bab-1=a-1, cac=a41, cbc=a28b3 >
Subgroups: 484 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, SD16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×D4, C22×C14, D4.4D4, C8×D7, C8⋊D7, C4.Dic7, C4.Dic7, D4⋊D7, D4.D7, C2×C56, C7×D8, C4○D28, D4×C14, C56.C4, C28.D4, D28.2C4, D4.D14, C14×D8, C56.23D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.4D4, D4×D7, D4⋊2D7, C2×C7⋊D4, C28⋊2D4, C56.23D4
►On 112 points
Generators in S112
(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 57 43 71 29 85 15 99)(2 112 44 70 30 84 16 98)(3 111 45 69 31 83 17 97)(4 110 46 68 32 82 18 96)(5 109 47 67 33 81 19 95)(6 108 48 66 34 80 20 94)(7 107 49 65 35 79 21 93)(8 106 50 64 36 78 22 92)(9 105 51 63 37 77 23 91)(10 104 52 62 38 76 24 90)(11 103 53 61 39 75 25 89)(12 102 54 60 40 74 26 88)(13 101 55 59 41 73 27 87)(14 100 56 58 42 72 28 86)
(1 57)(2 98)(3 83)(4 68)(5 109)(6 94)(7 79)(8 64)(9 105)(10 90)(11 75)(12 60)(13 101)(14 86)(15 71)(16 112)(17 97)(18 82)(19 67)(20 108)(21 93)(22 78)(23 63)(24 104)(25 89)(26 74)(27 59)(28 100)(29 85)(30 70)(31 111)(32 96)(33 81)(34 66)(35 107)(36 92)(37 77)(38 62)(39 103)(40 88)(41 73)(42 58)(43 99)(44 84)(45 69)(46 110)(47 95)(48 80)(49 65)(50 106)(51 91)(52 76)(53 61)(54 102)(55 87)(56 72)
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), (1,57,43,71,29,85,15,99)(2,112,44,70,30,84,16,98)(3,111,45,69,31,83,17,97)(4,110,46,68,32,82,18,96)(5,109,47,67,33,81,19,95)(6,108,48,66,34,80,20,94)(7,107,49,65,35,79,21,93)(8,106,50,64,36,78,22,92)(9,105,51,63,37,77,23,91)(10,104,52,62,38,76,24,90)(11,103,53,61,39,75,25,89)(12,102,54,60,40,74,26,88)(13,101,55,59,41,73,27,87)(14,100,56,58,42,72,28,86), (1,57)(2,98)(3,83)(4,68)(5,109)(6,94)(7,79)(8,64)(9,105)(10,90)(11,75)(12,60)(13,101)(14,86)(15,71)(16,112)(17,97)(18,82)(19,67)(20,108)(21,93)(22,78)(23,63)(24,104)(25,89)(26,74)(27,59)(28,100)(29,85)(30,70)(31,111)(32,96)(33,81)(34,66)(35,107)(36,92)(37,77)(38,62)(39,103)(40,88)(41,73)(42,58)(43,99)(44,84)(45,69)(46,110)(47,95)(48,80)(49,65)(50,106)(51,91)(52,76)(53,61)(54,102)(55,87)(56,72)>;
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), (1,57,43,71,29,85,15,99)(2,112,44,70,30,84,16,98)(3,111,45,69,31,83,17,97)(4,110,46,68,32,82,18,96)(5,109,47,67,33,81,19,95)(6,108,48,66,34,80,20,94)(7,107,49,65,35,79,21,93)(8,106,50,64,36,78,22,92)(9,105,51,63,37,77,23,91)(10,104,52,62,38,76,24,90)(11,103,53,61,39,75,25,89)(12,102,54,60,40,74,26,88)(13,101,55,59,41,73,27,87)(14,100,56,58,42,72,28,86), (1,57)(2,98)(3,83)(4,68)(5,109)(6,94)(7,79)(8,64)(9,105)(10,90)(11,75)(12,60)(13,101)(14,86)(15,71)(16,112)(17,97)(18,82)(19,67)(20,108)(21,93)(22,78)(23,63)(24,104)(25,89)(26,74)(27,59)(28,100)(29,85)(30,70)(31,111)(32,96)(33,81)(34,66)(35,107)(36,92)(37,77)(38,62)(39,103)(40,88)(41,73)(42,58)(43,99)(44,84)(45,69)(46,110)(47,95)(48,80)(49,65)(50,106)(51,91)(52,76)(53,61)(54,102)(55,87)(56,72) );
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)], [(1,57,43,71,29,85,15,99),(2,112,44,70,30,84,16,98),(3,111,45,69,31,83,17,97),(4,110,46,68,32,82,18,96),(5,109,47,67,33,81,19,95),(6,108,48,66,34,80,20,94),(7,107,49,65,35,79,21,93),(8,106,50,64,36,78,22,92),(9,105,51,63,37,77,23,91),(10,104,52,62,38,76,24,90),(11,103,53,61,39,75,25,89),(12,102,54,60,40,74,26,88),(13,101,55,59,41,73,27,87),(14,100,56,58,42,72,28,86)], [(1,57),(2,98),(3,83),(4,68),(5,109),(6,94),(7,79),(8,64),(9,105),(10,90),(11,75),(12,60),(13,101),(14,86),(15,71),(16,112),(17,97),(18,82),(19,67),(20,108),(21,93),(22,78),(23,63),(24,104),(25,89),(26,74),(27,59),(28,100),(29,85),(30,70),(31,111),(32,96),(33,81),(34,66),(35,107),(36,92),(37,77),(38,62),(39,103),(40,88),(41,73),(42,58),(43,99),(44,84),(45,69),(46,110),(47,95),(48,80),(49,65),(50,106),(51,91),(52,76),(53,61),(54,102),(55,87),(56,72)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 8 | 28 | 2 | 2 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 56 | 56 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D4.4D4 | D4×D7 | D4⋊2D7 | C56.23D4 |
| kernel | C56.23D4 | C56.C4 | C28.D4 | D28.2C4 | D4.D14 | C14×D8 | C56 | Dic14 | D28 | C2×D8 | C2×C14 | C2×C8 | C2×D4 | C8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56.23D4 ►in GL4(𝔽113) generated by
| 44 | 44 | 0 | 0 |
| 69 | 44 | 0 | 0 |
| 51 | 57 | 9 | 9 |
| 6 | 104 | 104 | 9 |
| 13 | 13 | 0 | 16 |
| 40 | 40 | 16 | 0 |
| 44 | 37 | 73 | 73 |
| 77 | 70 | 100 | 100 |
| 73 | 73 | 97 | 0 |
| 100 | 100 | 0 | 97 |
| 83 | 76 | 40 | 40 |
| 36 | 43 | 13 | 13 |
G:=sub<GL(4,GF(113))| [44,69,51,6,44,44,57,104,0,0,9,104,0,0,9,9],[13,40,44,77,13,40,37,70,0,16,73,100,16,0,73,100],[73,100,83,36,73,100,76,43,97,0,40,13,0,97,40,13] >;
C56.23D4 in GAP, Magma, Sage, TeX
C_{56}._{23}D_4 % in TeX
G:=Group("C56.23D4"); // GroupNames label
G:=SmallGroup(448,694);
// by ID
G=gap.SmallGroup(448,694);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,1123,297,136,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=c^2=1,b^4=a^28,b*a*b^-1=a^-1,c*a*c=a^41,c*b*c=a^28*b^3>;
// generators/relations