metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊2C4, C14.5D8, C2.2D56, C28.45D4, C14.3SD16, C22.10D28, (C2×C8)⋊2D7, (C2×C56)⋊2C2, C4.8(C4×D7), C4⋊Dic7⋊1C2, C7⋊2(D4⋊C4), C28.18(C2×C4), (C2×D28).1C2, (C2×C14).15D4, (C2×C4).71D14, C2.8(D14⋊C4), C2.3(C56⋊C2), C4.20(C7⋊D4), C14.7(C22⋊C4), (C2×C28).83C22, SmallGroup(224,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D56
G = < a,b,c | a2=b56=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
Smallest permutation representation of C2.D56
►On 112 points
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)
(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 82 83 56)(2 55 84 81)(3 80 85 54)(4 53 86 79)(5 78 87 52)(6 51 88 77)(7 76 89 50)(8 49 90 75)(9 74 91 48)(10 47 92 73)(11 72 93 46)(12 45 94 71)(13 70 95 44)(14 43 96 69)(15 68 97 42)(16 41 98 67)(17 66 99 40)(18 39 100 65)(19 64 101 38)(20 37 102 63)(21 62 103 36)(22 35 104 61)(23 60 105 34)(24 33 106 59)(25 58 107 32)(26 31 108 57)(27 112 109 30)(28 29 110 111)
G:=sub<Sym(112)| (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82), (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), (1,82,83,56)(2,55,84,81)(3,80,85,54)(4,53,86,79)(5,78,87,52)(6,51,88,77)(7,76,89,50)(8,49,90,75)(9,74,91,48)(10,47,92,73)(11,72,93,46)(12,45,94,71)(13,70,95,44)(14,43,96,69)(15,68,97,42)(16,41,98,67)(17,66,99,40)(18,39,100,65)(19,64,101,38)(20,37,102,63)(21,62,103,36)(22,35,104,61)(23,60,105,34)(24,33,106,59)(25,58,107,32)(26,31,108,57)(27,112,109,30)(28,29,110,111)>;
G:=Group( (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62)(37,63)(38,64)(39,65)(40,66)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82), (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), (1,82,83,56)(2,55,84,81)(3,80,85,54)(4,53,86,79)(5,78,87,52)(6,51,88,77)(7,76,89,50)(8,49,90,75)(9,74,91,48)(10,47,92,73)(11,72,93,46)(12,45,94,71)(13,70,95,44)(14,43,96,69)(15,68,97,42)(16,41,98,67)(17,66,99,40)(18,39,100,65)(19,64,101,38)(20,37,102,63)(21,62,103,36)(22,35,104,61)(23,60,105,34)(24,33,106,59)(25,58,107,32)(26,31,108,57)(27,112,109,30)(28,29,110,111) );
G=PermutationGroup([[(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,97),(16,98),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62),(37,63),(38,64),(39,65),(40,66),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82)], [(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)], [(1,82,83,56),(2,55,84,81),(3,80,85,54),(4,53,86,79),(5,78,87,52),(6,51,88,77),(7,76,89,50),(8,49,90,75),(9,74,91,48),(10,47,92,73),(11,72,93,46),(12,45,94,71),(13,70,95,44),(14,43,96,69),(15,68,97,42),(16,41,98,67),(17,66,99,40),(18,39,100,65),(19,64,101,38),(20,37,102,63),(21,62,103,36),(22,35,104,61),(23,60,105,34),(24,33,106,59),(25,58,107,32),(26,31,108,57),(27,112,109,30),(28,29,110,111)]])
C2.D56 is a maximal subgroup of
C4×C56⋊C2 C4×D56 C4.5D56 C42.264D14 C42.16D14 D56⋊C4 C42.19D14 C42.20D14 D28.31D4 D28⋊13D4 D28.32D4 D28⋊14D4 C23.38D28 C22.D56 C23.13D28 Dic7⋊4D8 Dic7.SD16 C4⋊C4.D14 D7×D4⋊C4 (D4×D7)⋊C4 D14⋊D8 C7⋊C8⋊D4 D4⋊D7⋊C4 Dic7⋊7SD16 Q8⋊C4⋊D7 Q8⋊Dic7⋊C2 Q8⋊(C4×D7) Q8⋊2D7⋊C4 D14⋊2SD16 C7⋊(C8⋊D4) Q8⋊D7⋊C4 D28⋊3Q8 C4⋊D56 D28.19D4 C42.36D14 D28⋊4Q8 D28.3Q8 Dic14⋊8D4 D14.4SD16 C4.Q8⋊D7 D28⋊Q8 D28.Q8 D14.5D8 C2.D8⋊D7 D28⋊2Q8 D28.2Q8 C23.23D28 C56⋊30D4 C56⋊29D4 C23.48D28 C23.49D28 C56⋊2D4 C56⋊3D4 Dic7⋊D8 D28⋊D4 Dic7⋊5SD16 (C7×D4).D4 D14⋊6SD16 D28⋊7D4 (C2×Q16)⋊D7 D28.17D4
C2.D56 is a maximal quotient of
C4.17D56 C22.2D56 C4.D56 C28.2D8 C56.78D4 C2.D112 D56.1C4 C28.3D8 C28.4D8 D56⋊2C4 C28.9C42
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4×D7 | C7⋊D4 | D28 | C56⋊C2 | D56 |
| kernel | C2.D56 | C4⋊Dic7 | C2×C56 | C2×D28 | D28 | C28 | C2×C14 | C2×C8 | C14 | C14 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of C2.D56 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 15 | 0 | 0 |
| 0 | 81 | 24 |
| 0 | 89 | 71 |
| 98 | 0 | 0 |
| 0 | 81 | 24 |
| 0 | 75 | 32 |
G:=sub<GL(3,GF(113))| [112,0,0,0,112,0,0,0,112],[15,0,0,0,81,89,0,24,71],[98,0,0,0,81,75,0,24,32] >;
C2.D56 in GAP, Magma, Sage, TeX
C_2.D_{56} % in TeX
G:=Group("C2.D56"); // GroupNames label
G:=SmallGroup(224,27);
// by ID
G=gap.SmallGroup(224,27);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,73,79,362,86,6917]);
// Polycyclic
G:=Group<a,b,c|a^2=b^56=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations
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