metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28:3D4, Dic7:1D4, C23.10D14, (C2xD4):6D7, (D4xC14):4C2, (C2xD28):9C2, C4:1(C7:D4), C7:2(C4:1D4), C2.28(D4xD7), (C4xDic7):6C2, C14.52(C2xD4), (C2xC4).52D14, (C2xC14).55C23, (C2xC28).35C22, C22.62(C22xD7), (C22xC14).22C22, (C2xDic7).39C22, (C22xD7).12C22, (C2xC7:D4):7C2, C2.16(C2xC7:D4), SmallGroup(224,135)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28:D4
G = < a,b,c | a28=b4=c2=1, bab-1=a13, cac=a-1, cbc=b-1 >
Subgroups: 510 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2xC4, C2xC4, D4, C23, C23, D7, C14, C14, C14, C42, C2xD4, C2xD4, Dic7, C28, D14, C2xC14, C2xC14, C4:1D4, D28, C2xDic7, C7:D4, C2xC28, C7xD4, C22xD7, C22xC14, C4xDic7, C2xD28, C2xC7:D4, D4xC14, C28:D4
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C4:1D4, C7:D4, C22xD7, D4xD7, C2xC7:D4, C28:D4
►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 62 95 50)(2 75 96 35)(3 60 97 48)(4 73 98 33)(5 58 99 46)(6 71 100 31)(7 84 101 44)(8 69 102 29)(9 82 103 42)(10 67 104 55)(11 80 105 40)(12 65 106 53)(13 78 107 38)(14 63 108 51)(15 76 109 36)(16 61 110 49)(17 74 111 34)(18 59 112 47)(19 72 85 32)(20 57 86 45)(21 70 87 30)(22 83 88 43)(23 68 89 56)(24 81 90 41)(25 66 91 54)(26 79 92 39)(27 64 93 52)(28 77 94 37)
(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 84)(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)
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), (1,62,95,50)(2,75,96,35)(3,60,97,48)(4,73,98,33)(5,58,99,46)(6,71,100,31)(7,84,101,44)(8,69,102,29)(9,82,103,42)(10,67,104,55)(11,80,105,40)(12,65,106,53)(13,78,107,38)(14,63,108,51)(15,76,109,36)(16,61,110,49)(17,74,111,34)(18,59,112,47)(19,72,85,32)(20,57,86,45)(21,70,87,30)(22,83,88,43)(23,68,89,56)(24,81,90,41)(25,66,91,54)(26,79,92,39)(27,64,93,52)(28,77,94,37), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84)(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)>;
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), (1,62,95,50)(2,75,96,35)(3,60,97,48)(4,73,98,33)(5,58,99,46)(6,71,100,31)(7,84,101,44)(8,69,102,29)(9,82,103,42)(10,67,104,55)(11,80,105,40)(12,65,106,53)(13,78,107,38)(14,63,108,51)(15,76,109,36)(16,61,110,49)(17,74,111,34)(18,59,112,47)(19,72,85,32)(20,57,86,45)(21,70,87,30)(22,83,88,43)(23,68,89,56)(24,81,90,41)(25,66,91,54)(26,79,92,39)(27,64,93,52)(28,77,94,37), (2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84)(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) );
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)], [(1,62,95,50),(2,75,96,35),(3,60,97,48),(4,73,98,33),(5,58,99,46),(6,71,100,31),(7,84,101,44),(8,69,102,29),(9,82,103,42),(10,67,104,55),(11,80,105,40),(12,65,106,53),(13,78,107,38),(14,63,108,51),(15,76,109,36),(16,61,110,49),(17,74,111,34),(18,59,112,47),(19,72,85,32),(20,57,86,45),(21,70,87,30),(22,83,88,43),(23,68,89,56),(24,81,90,41),(25,66,91,54),(26,79,92,39),(27,64,93,52),(28,77,94,37)], [(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(29,83),(30,82),(31,81),(32,80),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(56,84),(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)]])
C28:D4 is a maximal subgroup of
C23.2D28 D28:1D4 Dic7.SD16 Dic14:2D4 C4:C4.D14 D28:3D4 Dic7:D8 C56:5D4 C56:11D4 Dic7:5SD16 C56:15D4 C56:9D4 D28:18D4 2+ 1+4:D7 C42.228D14 Dic14:24D4 C42.114D14 C42.116D14 C24:3D14 C24.34D14 C24.36D14 C28:(C4oD4) Dic14:20D4 C14.382+ 1+4 D28:19D4 C14.442+ 1+4 C14.472+ 1+4 C14.482+ 1+4 C14.662+ 1+4 C14.672+ 1+4 C14.682+ 1+4 C42.233D14 C42:18D14 C42.143D14 D7xC4:1D4 C42:26D14 Dic14:11D4 D4xC7:D4 C24.41D14 C14.1462+ 1+4 (C2xC28):17D4 C14.1482+ 1+4
C28:D4 is a maximal quotient of
C24.7D14 C24.13D14 C23:2D28 (C4xDic7):8C4 (C2xD28):10C4 (C2xC28).289D4 C42.64D14 C42.214D14 C42.65D14 C28:D8 C42.74D14 C28:4SD16 C28:6SD16 C42.80D14 C28:3Q16 C56:5D4 C56:11D4 C56.22D4 C56.31D4 C56.43D4 C56:15D4 C56:9D4 C56.26D4 C56.37D4 C56.28D4 C24.19D14 C24.21D14
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 28 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7:D4 | D4xD7 |
| kernel | C28:D4 | C4xDic7 | C2xD28 | C2xC7:D4 | D4xC14 | Dic7 | C28 | C2xD4 | C2xC4 | C23 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 3 | 3 | 6 | 12 | 6 |
Matrix representation of C28:D4 ►in GL4(F29) generated by
| 1 | 26 | 0 | 0 |
| 3 | 21 | 0 | 0 |
| 0 | 0 | 28 | 17 |
| 0 | 0 | 5 | 1 |
| 5 | 16 | 0 | 0 |
| 2 | 24 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 24 | 28 |
G:=sub<GL(4,GF(29))| [1,3,0,0,26,21,0,0,0,0,28,5,0,0,17,1],[5,2,0,0,16,24,0,0,0,0,28,0,0,0,0,28],[1,3,0,0,0,28,0,0,0,0,1,24,0,0,0,28] >;
C28:D4 in GAP, Magma, Sage, TeX
C_{28}\rtimes D_4 % in TeX
G:=Group("C28:D4"); // GroupNames label
G:=SmallGroup(224,135);
// by ID
G=gap.SmallGroup(224,135);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,103,218,188,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations