metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D56⋊7C2, C4.20D28, C28.35D4, C8.17D14, Dic28⋊7C2, C22.1D28, C56.17C22, C28.30C23, D28.7C22, Dic14.6C22, (C2×C8)⋊4D7, (C2×C56)⋊6C2, C7⋊1(C4○D8), C4○D28⋊1C2, C56⋊C2⋊7C2, C2.13(C2×D28), C14.11(C2×D4), (C2×C14).18D4, (C2×C4).81D14, C4.28(C22×D7), (C2×C28).99C22, SmallGroup(224,99)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D56⋊7C2
G = < a,b,c | a56=b2=c2=1, bab=a-1, ac=ca, cbc=a28b >
Subgroups: 302 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C2×C8, D8, SD16, Q16, C4○D4, Dic7, C28, D14, C2×C14, C4○D8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C56⋊C2, D56, Dic28, C2×C56, C4○D28, D56⋊7C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4○D8, D28, C22×D7, C2×D28, D56⋊7C2
►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 7)(2 6)(3 5)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)
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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)>;
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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76) );
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)], [(1,7),(2,6),(3,5),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67),(76,112),(77,111),(78,110),(79,109),(80,108),(81,107),(82,106),(83,105),(84,104),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,109),(34,110),(35,111),(36,112),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76)]])
D56⋊7C2 is a maximal subgroup of
D56⋊8C4 Dic28.C4 D56.1C4 D56⋊2C4 D56⋊11C4 D56⋊4C4 D56⋊10C4 D56⋊7C4 D112⋊7C2 C16⋊D14 C16.D14 D8.D14 Q16.D14 C56.30C23 C56.9C23 D4.11D28 D4.12D28 D4.13D28 D8⋊13D14 D28.29D4 D28.30D4 D7×C4○D8 D8⋊10D14
D56⋊7C2 is a maximal quotient of
C56.13Q8 C4×C56⋊C2 C4×D56 C8.8D28 C42.264D14 C4×Dic28 C23.10D28 D28.32D4 D28⋊14D4 C23.13D28 Dic14.3Q8 D28.19D4 C42.36D14 D28.3Q8 C23.22D28 C23.23D28 C56⋊30D4 C56⋊29D4 C56.82D4
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 28 | 1 | 1 | 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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C4○D8 | D28 | D28 | D56⋊7C2 |
| kernel | D56⋊7C2 | C56⋊C2 | D56 | Dic28 | C2×C56 | C4○D28 | C28 | C2×C14 | C2×C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 6 | 3 | 4 | 6 | 6 | 24 |
Matrix representation of D56⋊7C2 ►in GL2(𝔽113) generated by
| 85 | 21 |
| 92 | 48 |
| 32 | 55 |
| 4 | 81 |
| 79 | 108 |
| 5 | 34 |
G:=sub<GL(2,GF(113))| [85,92,21,48],[32,4,55,81],[79,5,108,34] >;
D56⋊7C2 in GAP, Magma, Sage, TeX
D_{56}\rtimes_7C_2 % in TeX
G:=Group("D56:7C2"); // GroupNames label
G:=SmallGroup(224,99);
// by ID
G=gap.SmallGroup(224,99);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,50,579,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^56=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^28*b>;
// generators/relations