direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D56, C8⋊7D14, C14⋊1D8, C4.7D28, C56⋊8C22, C28.30D4, D28⋊3C22, C28.29C23, C22.13D28, C7⋊1(C2×D8), (C2×C8)⋊3D7, (C2×C56)⋊5C2, (C2×D28)⋊5C2, C2.12(C2×D28), C14.10(C2×D4), (C2×C4).80D14, (C2×C14).17D4, C4.27(C22×D7), (C2×C28).89C22, SmallGroup(224,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D56
G = < a,b,c | a2=b56=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 494 in 76 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, D4, C23, D7, C14, C14, C2×C8, D8, C2×D4, C28, D14, C2×C14, C2×D8, C56, D28, D28, C2×C28, C22×D7, D56, C2×C56, C2×D28, C2×D56
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, D28, C22×D7, D56, C2×D28, C2×D56
►On 112 points
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(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 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(50 56)(51 55)(52 54)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(106 112)(107 111)(108 110)
G:=sub<Sym(112)| (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(50,56)(51,55)(52,54)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(106,112)(107,111)(108,110)>;
G:=Group( (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(50,56)(51,55)(52,54)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(106,112)(107,111)(108,110) );
G=PermutationGroup([[(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)], [(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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(50,56),(51,55),(52,54),(57,105),(58,104),(59,103),(60,102),(61,101),(62,100),(63,99),(64,98),(65,97),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(106,112),(107,111),(108,110)]])
C2×D56 is a maximal subgroup of
C14.D16 D56.C4 C2.D112 C28.3D8 C28⋊4D8 C8.8D28 D56⋊C4 C8⋊D28 D28⋊13D4 D28⋊14D4 D4⋊D28 D28⋊3D4 D28⋊4D4 D28.12D4 C4⋊D56 D28.19D4 C56⋊7D4 D56⋊9C4 Dic7⋊5D8 C8⋊7D28 C8.21D28 C16⋊D14 C56⋊29D4 C56⋊3D4 D4.4D28 C56⋊5D4 C56⋊9D4 C56.28D4 Q16⋊D14 D4.12D28 C2×D7×D8 D8⋊15D14
C2×D56 is a maximal quotient of
C56⋊8Q8 C4.5D56 C28⋊4D8 D28⋊13D4 C22.D56 C4⋊D56 D28⋊4Q8 D112⋊7C2 C16⋊D14 C16.D14 C56⋊29D4
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D14 | D14 | D28 | D28 | D56 |
| kernel | C2×D56 | D56 | C2×C56 | C2×D28 | C28 | C2×C14 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 1 | 3 | 4 | 6 | 3 | 6 | 6 | 24 |
Matrix representation of C2×D56 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 112 | 0 | 0 |
| 0 | 101 | 84 |
| 0 | 29 | 70 |
| 112 | 0 | 0 |
| 0 | 79 | 112 |
| 0 | 25 | 34 |
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[112,0,0,0,101,29,0,84,70],[112,0,0,0,79,25,0,112,34] >;
C2×D56 in GAP, Magma, Sage, TeX
C_2\times D_{56} % in TeX
G:=Group("C2xD56"); // GroupNames label
G:=SmallGroup(224,98);
// by ID
G=gap.SmallGroup(224,98);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,122,579,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^2=b^56=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations