metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.C4, C8.12D14, Dic14.C4, M4(2)⋊5D7, C28.39C23, C56.12C22, C7⋊D4.C4, (C8×D7)⋊8C2, C7⋊2(C8○D4), C4.5(C4×D7), C8⋊D7⋊6C2, C28.13(C2×C4), C4○D28.3C2, D14.2(C2×C4), (C2×C4).46D14, C7⋊C8.12C22, C22.1(C4×D7), (C7×M4(2))⋊4C2, Dic7.4(C2×C4), C4.39(C22×D7), C14.16(C22×C4), (C2×C28).26C22, (C4×D7).16C22, (C2×C7⋊C8)⋊3C2, C2.17(C2×C4×D7), (C2×C14).6(C2×C4), SmallGroup(224,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.C4
G = < a,b,c | a28=b2=1, c4=a14, bab=a-1, cac-1=a15, cbc-1=a14b >
Subgroups: 206 in 62 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C2×C8, M4(2), 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, C2×C7⋊C8, C7×M4(2), C4○D28, D28.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, D14, C8○D4, C4×D7, C22×D7, C2×C4×D7, D28.C4
►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 59)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(104 112)(105 111)(106 110)(107 109)
(1 98 83 43 15 112 69 29)(2 85 84 30 16 99 70 44)(3 100 57 45 17 86 71 31)(4 87 58 32 18 101 72 46)(5 102 59 47 19 88 73 33)(6 89 60 34 20 103 74 48)(7 104 61 49 21 90 75 35)(8 91 62 36 22 105 76 50)(9 106 63 51 23 92 77 37)(10 93 64 38 24 107 78 52)(11 108 65 53 25 94 79 39)(12 95 66 40 26 109 80 54)(13 110 67 55 27 96 81 41)(14 97 68 42 28 111 82 56)
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)(57,59)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(104,112)(105,111)(106,110)(107,109), (1,98,83,43,15,112,69,29)(2,85,84,30,16,99,70,44)(3,100,57,45,17,86,71,31)(4,87,58,32,18,101,72,46)(5,102,59,47,19,88,73,33)(6,89,60,34,20,103,74,48)(7,104,61,49,21,90,75,35)(8,91,62,36,22,105,76,50)(9,106,63,51,23,92,77,37)(10,93,64,38,24,107,78,52)(11,108,65,53,25,94,79,39)(12,95,66,40,26,109,80,54)(13,110,67,55,27,96,81,41)(14,97,68,42,28,111,82,56)>;
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)(57,59)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(104,112)(105,111)(106,110)(107,109), (1,98,83,43,15,112,69,29)(2,85,84,30,16,99,70,44)(3,100,57,45,17,86,71,31)(4,87,58,32,18,101,72,46)(5,102,59,47,19,88,73,33)(6,89,60,34,20,103,74,48)(7,104,61,49,21,90,75,35)(8,91,62,36,22,105,76,50)(9,106,63,51,23,92,77,37)(10,93,64,38,24,107,78,52)(11,108,65,53,25,94,79,39)(12,95,66,40,26,109,80,54)(13,110,67,55,27,96,81,41)(14,97,68,42,28,111,82,56) );
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),(57,59),(60,84),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(104,112),(105,111),(106,110),(107,109)], [(1,98,83,43,15,112,69,29),(2,85,84,30,16,99,70,44),(3,100,57,45,17,86,71,31),(4,87,58,32,18,101,72,46),(5,102,59,47,19,88,73,33),(6,89,60,34,20,103,74,48),(7,104,61,49,21,90,75,35),(8,91,62,36,22,105,76,50),(9,106,63,51,23,92,77,37),(10,93,64,38,24,107,78,52),(11,108,65,53,25,94,79,39),(12,95,66,40,26,109,80,54),(13,110,67,55,27,96,81,41),(14,97,68,42,28,111,82,56)]])
D28.C4 is a maximal subgroup of
D28.2D4 D28.3D4 D28.6D4 D28.7D4 M4(2).22D14 C42.196D14 D56⋊10C4 D56⋊7C4 C28.70C24 D7×C8○D4 C56.49C23 D8⋊5D14 D8⋊6D14 C56.C23 D28.44D4
D28.C4 is a maximal quotient of
C56⋊Q8 C8⋊9D28 D14.4C42 C42.185D14 C56⋊C4⋊C2 C7⋊D4⋊C8 D14⋊C8⋊C2 C7⋊C8⋊26D4 Dic14⋊C8 C28.M4(2) D28⋊C8 C28⋊2M4(2) C42.30D14 C42.31D14 C28.439(C2×D4) C28.7C42 C56⋊D4 C56⋊18D4 (C2×D28).14C4
50 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 | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| 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 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 14 | 14 | 1 | 1 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | D14 | D14 | C8○D4 | C4×D7 | C4×D7 | D28.C4 |
| kernel | D28.C4 | C8×D7 | C8⋊D7 | C2×C7⋊C8 | C7×M4(2) | C4○D28 | Dic14 | D28 | C7⋊D4 | M4(2) | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 6 | 3 | 4 | 6 | 6 | 6 |
Matrix representation of D28.C4 ►in GL4(𝔽113) generated by
| 33 | 42 | 0 | 0 |
| 79 | 104 | 0 | 0 |
| 0 | 0 | 0 | 69 |
| 0 | 0 | 18 | 0 |
| 0 | 104 | 0 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 15 | 0 |
G:=sub<GL(4,GF(113))| [33,79,0,0,42,104,0,0,0,0,0,18,0,0,69,0],[0,25,0,0,104,0,0,0,0,0,1,0,0,0,0,112],[15,0,0,0,0,15,0,0,0,0,0,15,0,0,112,0] >;
D28.C4 in GAP, Magma, Sage, TeX
D_{28}.C_4 % in TeX
G:=Group("D28.C4"); // GroupNames label
G:=SmallGroup(224,102);
// by ID
G=gap.SmallGroup(224,102);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,188,50,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^2=1,c^4=a^14,b*a*b=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^14*b>;
// generators/relations