metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊10D4, C42⋊19D14, C14.1252+ 1+4, C4.70(D4×D7), (C4×D28)⋊43C2, C7⋊9(D4⋊5D4), (C2×Q8)⋊19D14, C28.63(C2×D4), C28⋊2D4⋊33C2, (C4×C28)⋊23C22, C22⋊C4⋊33D14, D14.24(C2×D4), C4.4D4⋊10D7, D14⋊11(C4○D4), C22⋊D28⋊24C2, D14⋊D4⋊40C2, D14⋊C4⋊54C22, D14⋊3Q8⋊28C2, (C2×D4).173D14, (C2×D28)⋊28C22, C4⋊Dic7⋊60C22, (Q8×C14)⋊13C22, C14.90(C22×D4), D14.D4⋊42C2, (C2×C28).601C23, (C2×C14).220C24, Dic7⋊C4⋊26C22, C2.49(D4⋊8D14), C23.D7⋊33C22, C23.42(C22×D7), (D4×C14).155C22, (C22×C14).50C23, (C23×D7).64C22, C22.241(C23×D7), (C2×Dic7).115C23, (C22×D7).215C23, (C2×D4×D7)⋊17C2, C2.63(C2×D4×D7), C2.76(D7×C4○D4), (C2×C4×D7)⋊26C22, (D7×C22⋊C4)⋊17C2, (C2×Q8⋊2D7)⋊11C2, C14.187(C2×C4○D4), (C7×C4.4D4)⋊12C2, (C2×C7⋊D4)⋊23C22, (C7×C22⋊C4)⋊29C22, (C2×C4).195(C22×D7), SmallGroup(448,1129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊10D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, ac=ca, dad=a13, cbc-1=a14b, dbd=a26b, dcd=c-1 >
Subgroups: 1932 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×D7, C22×C14, D4⋊5D4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, D4×D7, Q8⋊2D7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, C4×D28, D7×C22⋊C4, C22⋊D28, D14.D4, D14⋊D4, C28⋊2D4, D14⋊3Q8, C7×C4.4D4, C2×D4×D7, C2×Q8⋊2D7, D28⋊10D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C22×D7, D4⋊5D4, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, D4⋊8D14, D28⋊10D4
►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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)(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 91)(86 90)(87 89)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)
(1 83 92 52)(2 84 93 53)(3 57 94 54)(4 58 95 55)(5 59 96 56)(6 60 97 29)(7 61 98 30)(8 62 99 31)(9 63 100 32)(10 64 101 33)(11 65 102 34)(12 66 103 35)(13 67 104 36)(14 68 105 37)(15 69 106 38)(16 70 107 39)(17 71 108 40)(18 72 109 41)(19 73 110 42)(20 74 111 43)(21 75 112 44)(22 76 85 45)(23 77 86 46)(24 78 87 47)(25 79 88 48)(26 80 89 49)(27 81 90 50)(28 82 91 51)
(1 52)(2 37)(3 50)(4 35)(5 48)(6 33)(7 46)(8 31)(9 44)(10 29)(11 42)(12 55)(13 40)(14 53)(15 38)(16 51)(17 36)(18 49)(19 34)(20 47)(21 32)(22 45)(23 30)(24 43)(25 56)(26 41)(27 54)(28 39)(57 90)(58 103)(59 88)(60 101)(61 86)(62 99)(63 112)(64 97)(65 110)(66 95)(67 108)(68 93)(69 106)(70 91)(71 104)(72 89)(73 102)(74 87)(75 100)(76 85)(77 98)(78 111)(79 96)(80 109)(81 94)(82 107)(83 92)(84 105)
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(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)(85,91)(86,90)(87,89)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103), (1,83,92,52)(2,84,93,53)(3,57,94,54)(4,58,95,55)(5,59,96,56)(6,60,97,29)(7,61,98,30)(8,62,99,31)(9,63,100,32)(10,64,101,33)(11,65,102,34)(12,66,103,35)(13,67,104,36)(14,68,105,37)(15,69,106,38)(16,70,107,39)(17,71,108,40)(18,72,109,41)(19,73,110,42)(20,74,111,43)(21,75,112,44)(22,76,85,45)(23,77,86,46)(24,78,87,47)(25,79,88,48)(26,80,89,49)(27,81,90,50)(28,82,91,51), (1,52)(2,37)(3,50)(4,35)(5,48)(6,33)(7,46)(8,31)(9,44)(10,29)(11,42)(12,55)(13,40)(14,53)(15,38)(16,51)(17,36)(18,49)(19,34)(20,47)(21,32)(22,45)(23,30)(24,43)(25,56)(26,41)(27,54)(28,39)(57,90)(58,103)(59,88)(60,101)(61,86)(62,99)(63,112)(64,97)(65,110)(66,95)(67,108)(68,93)(69,106)(70,91)(71,104)(72,89)(73,102)(74,87)(75,100)(76,85)(77,98)(78,111)(79,96)(80,109)(81,94)(82,107)(83,92)(84,105)>;
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(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)(85,91)(86,90)(87,89)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103), (1,83,92,52)(2,84,93,53)(3,57,94,54)(4,58,95,55)(5,59,96,56)(6,60,97,29)(7,61,98,30)(8,62,99,31)(9,63,100,32)(10,64,101,33)(11,65,102,34)(12,66,103,35)(13,67,104,36)(14,68,105,37)(15,69,106,38)(16,70,107,39)(17,71,108,40)(18,72,109,41)(19,73,110,42)(20,74,111,43)(21,75,112,44)(22,76,85,45)(23,77,86,46)(24,78,87,47)(25,79,88,48)(26,80,89,49)(27,81,90,50)(28,82,91,51), (1,52)(2,37)(3,50)(4,35)(5,48)(6,33)(7,46)(8,31)(9,44)(10,29)(11,42)(12,55)(13,40)(14,53)(15,38)(16,51)(17,36)(18,49)(19,34)(20,47)(21,32)(22,45)(23,30)(24,43)(25,56)(26,41)(27,54)(28,39)(57,90)(58,103)(59,88)(60,101)(61,86)(62,99)(63,112)(64,97)(65,110)(66,95)(67,108)(68,93)(69,106)(70,91)(71,104)(72,89)(73,102)(74,87)(75,100)(76,85)(77,98)(78,111)(79,96)(80,109)(81,94)(82,107)(83,92)(84,105) );
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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,28),(23,27),(24,26),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(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),(85,91),(86,90),(87,89),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103)], [(1,83,92,52),(2,84,93,53),(3,57,94,54),(4,58,95,55),(5,59,96,56),(6,60,97,29),(7,61,98,30),(8,62,99,31),(9,63,100,32),(10,64,101,33),(11,65,102,34),(12,66,103,35),(13,67,104,36),(14,68,105,37),(15,69,106,38),(16,70,107,39),(17,71,108,40),(18,72,109,41),(19,73,110,42),(20,74,111,43),(21,75,112,44),(22,76,85,45),(23,77,86,46),(24,78,87,47),(25,79,88,48),(26,80,89,49),(27,81,90,50),(28,82,91,51)], [(1,52),(2,37),(3,50),(4,35),(5,48),(6,33),(7,46),(8,31),(9,44),(10,29),(11,42),(12,55),(13,40),(14,53),(15,38),(16,51),(17,36),(18,49),(19,34),(20,47),(21,32),(22,45),(23,30),(24,43),(25,56),(26,41),(27,54),(28,39),(57,90),(58,103),(59,88),(60,101),(61,86),(62,99),(63,112),(64,97),(65,110),(66,95),(67,108),(68,93),(69,106),(70,91),(71,104),(72,89),(73,102),(74,87),(75,100),(76,85),(77,98),(78,111),(79,96),(80,109),(81,94),(82,107),(83,92),(84,105)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 4 | 4 | 4 | 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 | 4 | 4 | 14 | ··· | 14 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 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 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4×D7 | D7×C4○D4 | D4⋊8D14 |
| kernel | D28⋊10D4 | C4×D28 | D7×C22⋊C4 | C22⋊D28 | D14.D4 | D14⋊D4 | C28⋊2D4 | D14⋊3Q8 | C7×C4.4D4 | C2×D4×D7 | C2×Q8⋊2D7 | D28 | C4.4D4 | D14 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 3 | 12 | 3 | 3 | 1 | 6 | 6 | 6 |
Matrix representation of D28⋊10D4 ►in GL6(𝔽29)
| 4 | 3 | 0 | 0 | 0 | 0 |
| 14 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 16 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 14 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 13 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 21 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [4,14,0,0,0,0,3,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,16,17],[4,14,0,0,0,0,1,25,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,13,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,11,28],[0,18,0,0,0,0,21,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,11,28] >;
D28⋊10D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_{10}D_4 % in TeX
G:=Group("D28:10D4"); // GroupNames label
G:=SmallGroup(448,1129);
// by ID
G=gap.SmallGroup(448,1129);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,1571,570,297,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^13,c*b*c^-1=a^14*b,d*b*d=a^26*b,d*c*d=c^-1>;
// generators/relations