metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.2C4, C8.18D14, C56.23C22, C28.37C23, Dic14.2C4, (C2×C8)⋊7D7, (C8×D7)⋊6C2, C7⋊1(C8○D4), (C2×C56)⋊10C2, C8⋊D7⋊7C2, C4.10(C4×D7), C7⋊D4.2C4, C28.20(C2×C4), C4○D28.6C2, D14.1(C2×C4), (C2×C4).78D14, C7⋊C8.11C22, C22.2(C4×D7), C4.Dic7⋊11C2, Dic7.3(C2×C4), C4.37(C22×D7), C14.14(C22×C4), (C2×C28).98C22, (C4×D7).15C22, C2.15(C2×C4×D7), (C2×C14).16(C2×C4), SmallGroup(224,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.2C4
G = < a,b,c | a28=b2=1, c4=a14, bab=a-1, ac=ca, bc=cb >
Subgroups: 206 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C2×C8, C2×C8, M4(2), C4○D4, Dic7, C28, D14, C2×C14, C8○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C8×D7, C8⋊D7, C4.Dic7, C2×C56, C4○D28, D28.2C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, D14, C8○D4, C4×D7, C22×D7, C2×C4×D7, D28.2C4
►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 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(50 56)(51 55)(52 54)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(106 112)(107 111)(108 110)
(1 106 36 67 15 92 50 81)(2 107 37 68 16 93 51 82)(3 108 38 69 17 94 52 83)(4 109 39 70 18 95 53 84)(5 110 40 71 19 96 54 57)(6 111 41 72 20 97 55 58)(7 112 42 73 21 98 56 59)(8 85 43 74 22 99 29 60)(9 86 44 75 23 100 30 61)(10 87 45 76 24 101 31 62)(11 88 46 77 25 102 32 63)(12 89 47 78 26 103 33 64)(13 90 48 79 27 104 34 65)(14 91 49 80 28 105 35 66)
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)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(50,56)(51,55)(52,54)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(106,112)(107,111)(108,110), (1,106,36,67,15,92,50,81)(2,107,37,68,16,93,51,82)(3,108,38,69,17,94,52,83)(4,109,39,70,18,95,53,84)(5,110,40,71,19,96,54,57)(6,111,41,72,20,97,55,58)(7,112,42,73,21,98,56,59)(8,85,43,74,22,99,29,60)(9,86,44,75,23,100,30,61)(10,87,45,76,24,101,31,62)(11,88,46,77,25,102,32,63)(12,89,47,78,26,103,33,64)(13,90,48,79,27,104,34,65)(14,91,49,80,28,105,35,66)>;
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)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(50,56)(51,55)(52,54)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(106,112)(107,111)(108,110), (1,106,36,67,15,92,50,81)(2,107,37,68,16,93,51,82)(3,108,38,69,17,94,52,83)(4,109,39,70,18,95,53,84)(5,110,40,71,19,96,54,57)(6,111,41,72,20,97,55,58)(7,112,42,73,21,98,56,59)(8,85,43,74,22,99,29,60)(9,86,44,75,23,100,30,61)(10,87,45,76,24,101,31,62)(11,88,46,77,25,102,32,63)(12,89,47,78,26,103,33,64)(13,90,48,79,27,104,34,65)(14,91,49,80,28,105,35,66) );
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),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(50,56),(51,55),(52,54),(57,83),(58,82),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(106,112),(107,111),(108,110)], [(1,106,36,67,15,92,50,81),(2,107,37,68,16,93,51,82),(3,108,38,69,17,94,52,83),(4,109,39,70,18,95,53,84),(5,110,40,71,19,96,54,57),(6,111,41,72,20,97,55,58),(7,112,42,73,21,98,56,59),(8,85,43,74,22,99,29,60),(9,86,44,75,23,100,30,61),(10,87,45,76,24,101,31,62),(11,88,46,77,25,102,32,63),(12,89,47,78,26,103,33,64),(13,90,48,79,27,104,34,65),(14,91,49,80,28,105,35,66)]])
D28.2C4 is a maximal subgroup of
D28.C8 Dic14.C8 D56⋊11C4 D56⋊4C4 C8.20D28 C8.21D28 C8.24D28 D28.4C8 C16.12D14 C56.93D4 C56.50D4 C56.23D4 C56.44D4 C56.29D4 C28.70C24 D7×C8○D4 C56.49C23 D8⋊13D14 D28.29D4 D28.30D4 D8⋊15D14 D8⋊11D14 D8.10D14
D28.2C4 is a maximal quotient of
C8×Dic14 C56⋊11Q8 C8×D28 C8⋊6D28 D14.C42 C42.243D14 C56⋊C4⋊C2 D14⋊C8⋊C2 D14⋊2M4(2) Dic7⋊M4(2) C42.27D14 D14⋊3M4(2) C42.30D14 C42.31D14 C28.12C42 Dic7⋊C8⋊C2 C8×C7⋊D4 (C22×C8)⋊D7 C56⋊32D4
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 14 | 14 | 1 | 1 | 2 | 14 | 14 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | D14 | D14 | C8○D4 | C4×D7 | C4×D7 | D28.2C4 |
| kernel | D28.2C4 | C8×D7 | C8⋊D7 | C4.Dic7 | C2×C56 | C4○D28 | Dic14 | D28 | C7⋊D4 | C2×C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 6 | 3 | 4 | 6 | 6 | 24 |
Matrix representation of D28.2C4 ►in GL4(𝔽113) generated by
| 89 | 79 | 0 | 0 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 98 | 28 |
| 0 | 0 | 0 | 15 |
| 0 | 34 | 0 | 0 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 7 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 95 | 0 |
| 0 | 0 | 0 | 95 |
G:=sub<GL(4,GF(113))| [89,10,0,0,79,0,0,0,0,0,98,0,0,0,28,15],[0,10,0,0,34,0,0,0,0,0,112,7,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,95,0,0,0,0,95] >;
D28.2C4 in GAP, Magma, Sage, TeX
D_{28}._2C_4 % in TeX
G:=Group("D28.2C4"); // GroupNames label
G:=SmallGroup(224,96);
// by ID
G=gap.SmallGroup(224,96);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,50,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^2=1,c^4=a^14,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations