metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28:21D4, C14.1172+ 1+4, C4:C4:10D14, C22:Q8:7D7, C7:7(D4:5D4), (C2xQ8):16D14, C4.111(D4xD7), C4:D28:25C2, D14.20(C2xD4), C28.234(C2xD4), D28:C4:25C2, C22:D28:16C2, (Q8xC14):7C22, D14:C4:20C22, (C22xD28):16C2, (C2xD28):25C22, (C2xC28).54C23, C22:C4.57D14, C14.76(C22xD4), D14.5D4:17C2, C28.23D4:12C2, (C2xC14).174C24, Dic7:C4:53C22, C22:2(Q8:2D7), (C4xDic7):28C22, (C22xC4).236D14, C2.34(D4:8D14), (C23xD7).52C22, C23.189(C22xD7), C22.195(C23xD7), (C22xC14).202C23, (C22xC28).254C22, (C2xDic7).233C23, (C22xD7).196C23, C23.D7.115C22, C2.49(C2xD4xD7), (C4xC7:D4):22C2, (D7xC22:C4):8C2, (C2xC4xD7):18C22, (C2xC14):7(C4oD4), (C7xC4:C4):19C22, (C2xQ8:2D7):7C2, (C7xC22:Q8):10C2, C14.114(C2xC4oD4), C2.17(C2xQ8:2D7), (C2xC4).47(C22xD7), (C2xC7:D4).122C22, (C7xC22:C4).29C22, SmallGroup(448,1083)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28:21D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=dad=a13, cbc-1=dbd=a26b, dcd=c-1 >
Subgroups: 1980 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, D7, C14, C14, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, Dic7, C28, C28, D14, D14, C2xC14, C2xC14, C2xC14, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C22xD4, C2xC4oD4, C4xD7, D28, D28, C2xDic7, C7:D4, C2xC28, C2xC28, C2xC28, C7xQ8, C22xD7, C22xD7, C22xD7, C22xC14, D4:5D4, C4xDic7, Dic7:C4, D14:C4, D14:C4, C23.D7, C7xC22:C4, C7xC4:C4, C7xC4:C4, C2xC4xD7, C2xC4xD7, C2xD28, C2xD28, C2xD28, Q8:2D7, C2xC7:D4, C22xC28, Q8xC14, C23xD7, D7xC22:C4, C22:D28, D28:C4, D14.5D4, C4:D28, C4:D28, C4xC7:D4, C28.23D4, C7xC22:Q8, C22xD28, C2xQ8:2D7, D28:21D4
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, C4oD4, C24, D14, C22xD4, C2xC4oD4, 2+ 1+4, C22xD7, D4:5D4, D4xD7, Q8:2D7, C23xD7, C2xD4xD7, C2xQ8:2D7, D4:8D14, D28:21D4
►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 35)(30 34)(31 33)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)(76 84)(77 83)(78 82)(79 81)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)
(1 50 96 70)(2 35 97 83)(3 48 98 68)(4 33 99 81)(5 46 100 66)(6 31 101 79)(7 44 102 64)(8 29 103 77)(9 42 104 62)(10 55 105 75)(11 40 106 60)(12 53 107 73)(13 38 108 58)(14 51 109 71)(15 36 110 84)(16 49 111 69)(17 34 112 82)(18 47 85 67)(19 32 86 80)(20 45 87 65)(21 30 88 78)(22 43 89 63)(23 56 90 76)(24 41 91 61)(25 54 92 74)(26 39 93 59)(27 52 94 72)(28 37 95 57)
(1 50)(2 35)(3 48)(4 33)(5 46)(6 31)(7 44)(8 29)(9 42)(10 55)(11 40)(12 53)(13 38)(14 51)(15 36)(16 49)(17 34)(18 47)(19 32)(20 45)(21 30)(22 43)(23 56)(24 41)(25 54)(26 39)(27 52)(28 37)(57 95)(58 108)(59 93)(60 106)(61 91)(62 104)(63 89)(64 102)(65 87)(66 100)(67 85)(68 98)(69 111)(70 96)(71 109)(72 94)(73 107)(74 92)(75 105)(76 90)(77 103)(78 88)(79 101)(80 86)(81 99)(82 112)(83 97)(84 110)
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,35)(30,34)(31,33)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(76,84)(77,83)(78,82)(79,81)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100), (1,50,96,70)(2,35,97,83)(3,48,98,68)(4,33,99,81)(5,46,100,66)(6,31,101,79)(7,44,102,64)(8,29,103,77)(9,42,104,62)(10,55,105,75)(11,40,106,60)(12,53,107,73)(13,38,108,58)(14,51,109,71)(15,36,110,84)(16,49,111,69)(17,34,112,82)(18,47,85,67)(19,32,86,80)(20,45,87,65)(21,30,88,78)(22,43,89,63)(23,56,90,76)(24,41,91,61)(25,54,92,74)(26,39,93,59)(27,52,94,72)(28,37,95,57), (1,50)(2,35)(3,48)(4,33)(5,46)(6,31)(7,44)(8,29)(9,42)(10,55)(11,40)(12,53)(13,38)(14,51)(15,36)(16,49)(17,34)(18,47)(19,32)(20,45)(21,30)(22,43)(23,56)(24,41)(25,54)(26,39)(27,52)(28,37)(57,95)(58,108)(59,93)(60,106)(61,91)(62,104)(63,89)(64,102)(65,87)(66,100)(67,85)(68,98)(69,111)(70,96)(71,109)(72,94)(73,107)(74,92)(75,105)(76,90)(77,103)(78,88)(79,101)(80,86)(81,99)(82,112)(83,97)(84,110)>;
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,35)(30,34)(31,33)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(76,84)(77,83)(78,82)(79,81)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100), (1,50,96,70)(2,35,97,83)(3,48,98,68)(4,33,99,81)(5,46,100,66)(6,31,101,79)(7,44,102,64)(8,29,103,77)(9,42,104,62)(10,55,105,75)(11,40,106,60)(12,53,107,73)(13,38,108,58)(14,51,109,71)(15,36,110,84)(16,49,111,69)(17,34,112,82)(18,47,85,67)(19,32,86,80)(20,45,87,65)(21,30,88,78)(22,43,89,63)(23,56,90,76)(24,41,91,61)(25,54,92,74)(26,39,93,59)(27,52,94,72)(28,37,95,57), (1,50)(2,35)(3,48)(4,33)(5,46)(6,31)(7,44)(8,29)(9,42)(10,55)(11,40)(12,53)(13,38)(14,51)(15,36)(16,49)(17,34)(18,47)(19,32)(20,45)(21,30)(22,43)(23,56)(24,41)(25,54)(26,39)(27,52)(28,37)(57,95)(58,108)(59,93)(60,106)(61,91)(62,104)(63,89)(64,102)(65,87)(66,100)(67,85)(68,98)(69,111)(70,96)(71,109)(72,94)(73,107)(74,92)(75,105)(76,90)(77,103)(78,88)(79,101)(80,86)(81,99)(82,112)(83,97)(84,110) );
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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,35),(30,34),(31,33),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67),(76,84),(77,83),(78,82),(79,81),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100)], [(1,50,96,70),(2,35,97,83),(3,48,98,68),(4,33,99,81),(5,46,100,66),(6,31,101,79),(7,44,102,64),(8,29,103,77),(9,42,104,62),(10,55,105,75),(11,40,106,60),(12,53,107,73),(13,38,108,58),(14,51,109,71),(15,36,110,84),(16,49,111,69),(17,34,112,82),(18,47,85,67),(19,32,86,80),(20,45,87,65),(21,30,88,78),(22,43,89,63),(23,56,90,76),(24,41,91,61),(25,54,92,74),(26,39,93,59),(27,52,94,72),(28,37,95,57)], [(1,50),(2,35),(3,48),(4,33),(5,46),(6,31),(7,44),(8,29),(9,42),(10,55),(11,40),(12,53),(13,38),(14,51),(15,36),(16,49),(17,34),(18,47),(19,32),(20,45),(21,30),(22,43),(23,56),(24,41),(25,54),(26,39),(27,52),(28,37),(57,95),(58,108),(59,93),(60,106),(61,91),(62,104),(63,89),(64,102),(65,87),(66,100),(67,85),(68,98),(69,111),(70,96),(71,109),(72,94),(73,107),(74,92),(75,105),(76,90),(77,103),(78,88),(79,101),(80,86),(81,99),(82,112),(83,97),(84,110)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4oD4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4xD7 | Q8:2D7 | D4:8D14 |
| kernel | D28:21D4 | D7xC22:C4 | C22:D28 | D28:C4 | D14.5D4 | C4:D28 | C4xC7:D4 | C28.23D4 | C7xC22:Q8 | C22xD28 | C2xQ8:2D7 | D28 | C22:Q8 | C2xC14 | C22:C4 | C4:C4 | C22xC4 | C2xQ8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
Matrix representation of D28:21D4 ►in GL6(F29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 16 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 12 | 21 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 16 | 17 |
| 17 | 8 | 0 | 0 | 0 | 0 |
| 22 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 16 | 17 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,22,0,0,0,0,7,1,0,0,0,0,0,0,28,18,0,0,0,0,16,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,10,0,0,0,0,22,10,0,0,0,0,0,0,28,18,0,0,0,0,0,1],[12,0,0,0,0,0,21,17,0,0,0,0,0,0,28,22,0,0,0,0,0,1,0,0,0,0,0,0,12,16,0,0,0,0,11,17],[17,22,0,0,0,0,8,12,0,0,0,0,0,0,28,22,0,0,0,0,0,1,0,0,0,0,0,0,12,16,0,0,0,0,11,17] >;
D28:21D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_{21}D_4 % in TeX
G:=Group("D28:21D4"); // GroupNames label
G:=SmallGroup(448,1083);
// by ID
G=gap.SmallGroup(448,1083);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,297,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^13,c*b*c^-1=d*b*d=a^26*b,d*c*d=c^-1>;
// generators/relations