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), D14⋊D4⋊40C2, C22⋊D28⋊24C2, 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
Subgroups: 1932 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×29], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], D7 [×7], C14 [×3], C14 [×2], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×6], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×4], C28 [×2], C28 [×4], D14 [×6], D14 [×17], C2×C14, C2×C14 [×6], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×D7 [×10], D28 [×4], D28 [×6], C2×Dic7 [×2], C2×Dic7 [×2], C7⋊D4 [×6], C2×C28 [×3], C2×C28 [×2], C7×D4 [×2], C7×Q8 [×2], C22×D7 [×2], C22×D7 [×2], C22×D7 [×10], C22×C14 [×2], D4⋊5D4, Dic7⋊C4 [×2], C4⋊Dic7 [×2], D14⋊C4 [×2], D14⋊C4 [×4], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×4], C2×C4×D7 [×2], C2×C4×D7 [×4], C2×D28 [×2], C2×D28 [×2], D4×D7 [×4], Q8⋊2D7 [×4], C2×C7⋊D4 [×4], D4×C14, Q8×C14, C23×D7 [×2], C4×D28 [×2], D7×C22⋊C4 [×2], C22⋊D28 [×2], D14.D4 [×2], D14⋊D4 [×2], C28⋊2D4, D14⋊3Q8, C7×C4.4D4, C2×D4×D7, C2×Q8⋊2D7, D28⋊10D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊5D4, D4×D7 [×2], C23×D7, C2×D4×D7, D7×C4○D4, D4⋊8D14, D28⋊10D4
Generators and relations
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 >
►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 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(46 56)(47 55)(48 54)(49 53)(50 52)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(108 112)(109 111)
(1 82 86 34)(2 83 87 35)(3 84 88 36)(4 57 89 37)(5 58 90 38)(6 59 91 39)(7 60 92 40)(8 61 93 41)(9 62 94 42)(10 63 95 43)(11 64 96 44)(12 65 97 45)(13 66 98 46)(14 67 99 47)(15 68 100 48)(16 69 101 49)(17 70 102 50)(18 71 103 51)(19 72 104 52)(20 73 105 53)(21 74 106 54)(22 75 107 55)(23 76 108 56)(24 77 109 29)(25 78 110 30)(26 79 111 31)(27 80 112 32)(28 81 85 33)
(1 34)(2 47)(3 32)(4 45)(5 30)(6 43)(7 56)(8 41)(9 54)(10 39)(11 52)(12 37)(13 50)(14 35)(15 48)(16 33)(17 46)(18 31)(19 44)(20 29)(21 42)(22 55)(23 40)(24 53)(25 38)(26 51)(27 36)(28 49)(57 97)(58 110)(59 95)(60 108)(61 93)(62 106)(63 91)(64 104)(65 89)(66 102)(67 87)(68 100)(69 85)(70 98)(71 111)(72 96)(73 109)(74 94)(75 107)(76 92)(77 105)(78 90)(79 103)(80 88)(81 101)(82 86)(83 99)(84 112)
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)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111), (1,82,86,34)(2,83,87,35)(3,84,88,36)(4,57,89,37)(5,58,90,38)(6,59,91,39)(7,60,92,40)(8,61,93,41)(9,62,94,42)(10,63,95,43)(11,64,96,44)(12,65,97,45)(13,66,98,46)(14,67,99,47)(15,68,100,48)(16,69,101,49)(17,70,102,50)(18,71,103,51)(19,72,104,52)(20,73,105,53)(21,74,106,54)(22,75,107,55)(23,76,108,56)(24,77,109,29)(25,78,110,30)(26,79,111,31)(27,80,112,32)(28,81,85,33), (1,34)(2,47)(3,32)(4,45)(5,30)(6,43)(7,56)(8,41)(9,54)(10,39)(11,52)(12,37)(13,50)(14,35)(15,48)(16,33)(17,46)(18,31)(19,44)(20,29)(21,42)(22,55)(23,40)(24,53)(25,38)(26,51)(27,36)(28,49)(57,97)(58,110)(59,95)(60,108)(61,93)(62,106)(63,91)(64,104)(65,89)(66,102)(67,87)(68,100)(69,85)(70,98)(71,111)(72,96)(73,109)(74,94)(75,107)(76,92)(77,105)(78,90)(79,103)(80,88)(81,101)(82,86)(83,99)(84,112)>;
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)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,56)(47,55)(48,54)(49,53)(50,52)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111), (1,82,86,34)(2,83,87,35)(3,84,88,36)(4,57,89,37)(5,58,90,38)(6,59,91,39)(7,60,92,40)(8,61,93,41)(9,62,94,42)(10,63,95,43)(11,64,96,44)(12,65,97,45)(13,66,98,46)(14,67,99,47)(15,68,100,48)(16,69,101,49)(17,70,102,50)(18,71,103,51)(19,72,104,52)(20,73,105,53)(21,74,106,54)(22,75,107,55)(23,76,108,56)(24,77,109,29)(25,78,110,30)(26,79,111,31)(27,80,112,32)(28,81,85,33), (1,34)(2,47)(3,32)(4,45)(5,30)(6,43)(7,56)(8,41)(9,54)(10,39)(11,52)(12,37)(13,50)(14,35)(15,48)(16,33)(17,46)(18,31)(19,44)(20,29)(21,42)(22,55)(23,40)(24,53)(25,38)(26,51)(27,36)(28,49)(57,97)(58,110)(59,95)(60,108)(61,93)(62,106)(63,91)(64,104)(65,89)(66,102)(67,87)(68,100)(69,85)(70,98)(71,111)(72,96)(73,109)(74,94)(75,107)(76,92)(77,105)(78,90)(79,103)(80,88)(81,101)(82,86)(83,99)(84,112) );
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),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(46,56),(47,55),(48,54),(49,53),(50,52),(58,84),(59,83),(60,82),(61,81),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(85,107),(86,106),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(108,112),(109,111)], [(1,82,86,34),(2,83,87,35),(3,84,88,36),(4,57,89,37),(5,58,90,38),(6,59,91,39),(7,60,92,40),(8,61,93,41),(9,62,94,42),(10,63,95,43),(11,64,96,44),(12,65,97,45),(13,66,98,46),(14,67,99,47),(15,68,100,48),(16,69,101,49),(17,70,102,50),(18,71,103,51),(19,72,104,52),(20,73,105,53),(21,74,106,54),(22,75,107,55),(23,76,108,56),(24,77,109,29),(25,78,110,30),(26,79,111,31),(27,80,112,32),(28,81,85,33)], [(1,34),(2,47),(3,32),(4,45),(5,30),(6,43),(7,56),(8,41),(9,54),(10,39),(11,52),(12,37),(13,50),(14,35),(15,48),(16,33),(17,46),(18,31),(19,44),(20,29),(21,42),(22,55),(23,40),(24,53),(25,38),(26,51),(27,36),(28,49),(57,97),(58,110),(59,95),(60,108),(61,93),(62,106),(63,91),(64,104),(65,89),(66,102),(67,87),(68,100),(69,85),(70,98),(71,111),(72,96),(73,109),(74,94),(75,107),(76,92),(77,105),(78,90),(79,103),(80,88),(81,101),(82,86),(83,99),(84,112)])
Matrix representation ►G ⊆ 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] >;
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 |
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