metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊5D28, C42⋊15D14, C14.172+ 1+4, C4⋊C4⋊48D14, (C7×D4)⋊10D4, (C4×D4)⋊16D7, (D4×C28)⋊18C2, (C4×D28)⋊30C2, C28⋊7D4⋊9C2, C7⋊3(D4⋊5D4), C4.22(C2×D28), C28.54(C2×D4), D14⋊7(C4○D4), C22⋊D28⋊6C2, C4⋊D28⋊15C2, (C4×C28)⋊20C22, C22⋊C4⋊47D14, (C2×D28)⋊6C22, (C22×C4)⋊13D14, C4⋊Dic7⋊8C22, C22.1(C2×D28), D14⋊C4⋊52C22, D14⋊2Q8⋊14C2, (C2×D4).248D14, C4.D28⋊18C2, (C2×C14).98C24, C2.18(C22×D28), C14.16(C22×D4), (C2×C28).159C23, (C22×C28)⋊10C22, C22.D28⋊5C2, C2.18(D4⋊6D14), (C2×Dic14)⋊17C22, (D4×C14).259C22, (C2×Dic7).42C23, (C22×Dic7)⋊9C22, (C22×D7).33C23, (C23×D7).40C22, C22.123(C23×D7), C23.172(C22×D7), (C22×C14).168C23, (C2×D4×D7)⋊4C2, (C2×C4×D7)⋊3C22, C2.22(D7×C4○D4), (C2×C14).1(C2×D4), (C2×D14⋊C4)⋊21C2, (C2×D4⋊2D7)⋊3C2, (C7×C4⋊C4)⋊60C22, (C2×C7⋊D4)⋊4C22, C14.139(C2×C4○D4), (C7×C22⋊C4)⋊50C22, (C2×C4).160(C22×D7), SmallGroup(448,1007)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊5D28
G = < a,b,c,d | a4=b2=c28=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1908 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, D4⋊5D4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, D14⋊C4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×D28, D4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C4×D28, C4.D28, C22⋊D28, C22.D28, C4⋊D28, D14⋊2Q8, C2×D14⋊C4, C28⋊7D4, D4×C28, C2×D4×D7, C2×D4⋊2D7, D4⋊5D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, D28, C22×D7, D4⋊5D4, C2×D28, C23×D7, C22×D28, D4⋊6D14, D7×C4○D4, D4⋊5D28
►On 112 points
(1 52 108 77)(2 78 109 53)(3 54 110 79)(4 80 111 55)(5 56 112 81)(6 82 85 29)(7 30 86 83)(8 84 87 31)(9 32 88 57)(10 58 89 33)(11 34 90 59)(12 60 91 35)(13 36 92 61)(14 62 93 37)(15 38 94 63)(16 64 95 39)(17 40 96 65)(18 66 97 41)(19 42 98 67)(20 68 99 43)(21 44 100 69)(22 70 101 45)(23 46 102 71)(24 72 103 47)(25 48 104 73)(26 74 105 49)(27 50 106 75)(28 76 107 51)
(1 38)(2 64)(3 40)(4 66)(5 42)(6 68)(7 44)(8 70)(9 46)(10 72)(11 48)(12 74)(13 50)(14 76)(15 52)(16 78)(17 54)(18 80)(19 56)(20 82)(21 30)(22 84)(23 32)(24 58)(25 34)(26 60)(27 36)(28 62)(29 99)(31 101)(33 103)(35 105)(37 107)(39 109)(41 111)(43 85)(45 87)(47 89)(49 91)(51 93)(53 95)(55 97)(57 102)(59 104)(61 106)(63 108)(65 110)(67 112)(69 86)(71 88)(73 90)(75 92)(77 94)(79 96)(81 98)(83 100)
(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)(2 61)(3 60)(4 59)(5 58)(6 57)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 88)(30 87)(31 86)(32 85)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)
G:=sub<Sym(112)| (1,52,108,77)(2,78,109,53)(3,54,110,79)(4,80,111,55)(5,56,112,81)(6,82,85,29)(7,30,86,83)(8,84,87,31)(9,32,88,57)(10,58,89,33)(11,34,90,59)(12,60,91,35)(13,36,92,61)(14,62,93,37)(15,38,94,63)(16,64,95,39)(17,40,96,65)(18,66,97,41)(19,42,98,67)(20,68,99,43)(21,44,100,69)(22,70,101,45)(23,46,102,71)(24,72,103,47)(25,48,104,73)(26,74,105,49)(27,50,106,75)(28,76,107,51), (1,38)(2,64)(3,40)(4,66)(5,42)(6,68)(7,44)(8,70)(9,46)(10,72)(11,48)(12,74)(13,50)(14,76)(15,52)(16,78)(17,54)(18,80)(19,56)(20,82)(21,30)(22,84)(23,32)(24,58)(25,34)(26,60)(27,36)(28,62)(29,99)(31,101)(33,103)(35,105)(37,107)(39,109)(41,111)(43,85)(45,87)(47,89)(49,91)(51,93)(53,95)(55,97)(57,102)(59,104)(61,106)(63,108)(65,110)(67,112)(69,86)(71,88)(73,90)(75,92)(77,94)(79,96)(81,98)(83,100), (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)(2,61)(3,60)(4,59)(5,58)(6,57)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,88)(30,87)(31,86)(32,85)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)>;
G:=Group( (1,52,108,77)(2,78,109,53)(3,54,110,79)(4,80,111,55)(5,56,112,81)(6,82,85,29)(7,30,86,83)(8,84,87,31)(9,32,88,57)(10,58,89,33)(11,34,90,59)(12,60,91,35)(13,36,92,61)(14,62,93,37)(15,38,94,63)(16,64,95,39)(17,40,96,65)(18,66,97,41)(19,42,98,67)(20,68,99,43)(21,44,100,69)(22,70,101,45)(23,46,102,71)(24,72,103,47)(25,48,104,73)(26,74,105,49)(27,50,106,75)(28,76,107,51), (1,38)(2,64)(3,40)(4,66)(5,42)(6,68)(7,44)(8,70)(9,46)(10,72)(11,48)(12,74)(13,50)(14,76)(15,52)(16,78)(17,54)(18,80)(19,56)(20,82)(21,30)(22,84)(23,32)(24,58)(25,34)(26,60)(27,36)(28,62)(29,99)(31,101)(33,103)(35,105)(37,107)(39,109)(41,111)(43,85)(45,87)(47,89)(49,91)(51,93)(53,95)(55,97)(57,102)(59,104)(61,106)(63,108)(65,110)(67,112)(69,86)(71,88)(73,90)(75,92)(77,94)(79,96)(81,98)(83,100), (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)(2,61)(3,60)(4,59)(5,58)(6,57)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,88)(30,87)(31,86)(32,85)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89) );
G=PermutationGroup([[(1,52,108,77),(2,78,109,53),(3,54,110,79),(4,80,111,55),(5,56,112,81),(6,82,85,29),(7,30,86,83),(8,84,87,31),(9,32,88,57),(10,58,89,33),(11,34,90,59),(12,60,91,35),(13,36,92,61),(14,62,93,37),(15,38,94,63),(16,64,95,39),(17,40,96,65),(18,66,97,41),(19,42,98,67),(20,68,99,43),(21,44,100,69),(22,70,101,45),(23,46,102,71),(24,72,103,47),(25,48,104,73),(26,74,105,49),(27,50,106,75),(28,76,107,51)], [(1,38),(2,64),(3,40),(4,66),(5,42),(6,68),(7,44),(8,70),(9,46),(10,72),(11,48),(12,74),(13,50),(14,76),(15,52),(16,78),(17,54),(18,80),(19,56),(20,82),(21,30),(22,84),(23,32),(24,58),(25,34),(26,60),(27,36),(28,62),(29,99),(31,101),(33,103),(35,105),(37,107),(39,109),(41,111),(43,85),(45,87),(47,89),(49,91),(51,93),(53,95),(55,97),(57,102),(59,104),(61,106),(63,108),(65,110),(67,112),(69,86),(71,88),(73,90),(75,92),(77,94),(79,96),(81,98),(83,100)], [(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),(2,61),(3,60),(4,59),(5,58),(6,57),(7,84),(8,83),(9,82),(10,81),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,88),(30,87),(31,86),(32,85),(33,112),(34,111),(35,110),(36,109),(37,108),(38,107),(39,106),(40,105),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D14 | D28 | 2+ 1+4 | D4⋊6D14 | D7×C4○D4 |
| kernel | D4⋊5D28 | C4×D28 | C4.D28 | C22⋊D28 | C22.D28 | C4⋊D28 | D14⋊2Q8 | C2×D14⋊C4 | C28⋊7D4 | D4×C28 | C2×D4×D7 | C2×D4⋊2D7 | C7×D4 | C4×D4 | D14 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
Matrix representation of D4⋊5D28 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 17 | 24 | 0 | 0 |
| 5 | 2 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 5 | 0 | 0 |
| 12 | 17 | 0 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0],[28,0,0,0,0,28,0,0,0,0,0,1,0,0,1,0],[17,5,0,0,24,2,0,0,0,0,17,0,0,0,0,12],[12,12,0,0,5,17,0,0,0,0,0,12,0,0,17,0] >;
D4⋊5D28 in GAP, Magma, Sage, TeX
D_4\rtimes_5D_{28} % in TeX
G:=Group("D4:5D28"); // GroupNames label
G:=SmallGroup(448,1007);
// by ID
G=gap.SmallGroup(448,1007);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^28=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations