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), C28.54(C2×D4), C4.22(C2×D28), D14⋊7(C4○D4), C22⋊D28⋊6C2, C28⋊1D4⋊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, C14.16(C22×D4), C2.18(C22×D28), (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, (C23×D7).40C22, (C22×D7).33C23, C23.172(C22×D7), C22.123(C23×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
Subgroups: 1908 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×25], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×14], D7 [×5], C14 [×3], C14 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic7 [×4], C28 [×2], C28 [×4], D14 [×2], D14 [×19], C2×C14, C2×C14 [×4], C2×C14 [×4], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic14 [×2], C4×D7 [×4], D28 [×6], C2×Dic7 [×2], C2×Dic7 [×2], C2×Dic7 [×4], C7⋊D4 [×8], C2×C28 [×3], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C22×D7 [×2], C22×D7 [×2], C22×D7 [×10], C22×C14 [×2], D4⋊5D4, C4⋊Dic7, C4⋊Dic7 [×2], D14⋊C4 [×2], D14⋊C4 [×8], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×C4×D7 [×2], C2×D28 [×2], C2×D28 [×2], D4×D7 [×4], D4⋊2D7 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×4], C22×C28 [×2], D4×C14, C23×D7 [×2], C4×D28, C4.D28, C22⋊D28 [×2], C22.D28 [×2], C28⋊1D4, D14⋊2Q8, C2×D14⋊C4 [×2], C28⋊7D4 [×2], D4×C28, C2×D4×D7, C2×D4⋊2D7, D4⋊5D28
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), D28 [×4], C22×D7 [×7], D4⋊5D4, C2×D28 [×6], C23×D7, C22×D28, D4⋊6D14, D7×C4○D4, D4⋊5D28
Generators and relations
G = < a,b,c,d | a4=b2=c28=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 112 points
(1 84 45 88)(2 89 46 57)(3 58 47 90)(4 91 48 59)(5 60 49 92)(6 93 50 61)(7 62 51 94)(8 95 52 63)(9 64 53 96)(10 97 54 65)(11 66 55 98)(12 99 56 67)(13 68 29 100)(14 101 30 69)(15 70 31 102)(16 103 32 71)(17 72 33 104)(18 105 34 73)(19 74 35 106)(20 107 36 75)(21 76 37 108)(22 109 38 77)(23 78 39 110)(24 111 40 79)(25 80 41 112)(26 85 42 81)(27 82 43 86)(28 87 44 83)
(1 70)(2 103)(3 72)(4 105)(5 74)(6 107)(7 76)(8 109)(9 78)(10 111)(11 80)(12 85)(13 82)(14 87)(15 84)(16 89)(17 58)(18 91)(19 60)(20 93)(21 62)(22 95)(23 64)(24 97)(25 66)(26 99)(27 68)(28 101)(29 86)(30 83)(31 88)(32 57)(33 90)(34 59)(35 92)(36 61)(37 94)(38 63)(39 96)(40 65)(41 98)(42 67)(43 100)(44 69)(45 102)(46 71)(47 104)(48 73)(49 106)(50 75)(51 108)(52 77)(53 110)(54 79)(55 112)(56 81)
(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)
(1 101)(2 100)(3 99)(4 98)(5 97)(6 96)(7 95)(8 94)(9 93)(10 92)(11 91)(12 90)(13 89)(14 88)(15 87)(16 86)(17 85)(18 112)(19 111)(20 110)(21 109)(22 108)(23 107)(24 106)(25 105)(26 104)(27 103)(28 102)(29 57)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 77)(38 76)(39 75)(40 74)(41 73)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)
G:=sub<Sym(112)| (1,84,45,88)(2,89,46,57)(3,58,47,90)(4,91,48,59)(5,60,49,92)(6,93,50,61)(7,62,51,94)(8,95,52,63)(9,64,53,96)(10,97,54,65)(11,66,55,98)(12,99,56,67)(13,68,29,100)(14,101,30,69)(15,70,31,102)(16,103,32,71)(17,72,33,104)(18,105,34,73)(19,74,35,106)(20,107,36,75)(21,76,37,108)(22,109,38,77)(23,78,39,110)(24,111,40,79)(25,80,41,112)(26,85,42,81)(27,82,43,86)(28,87,44,83), (1,70)(2,103)(3,72)(4,105)(5,74)(6,107)(7,76)(8,109)(9,78)(10,111)(11,80)(12,85)(13,82)(14,87)(15,84)(16,89)(17,58)(18,91)(19,60)(20,93)(21,62)(22,95)(23,64)(24,97)(25,66)(26,99)(27,68)(28,101)(29,86)(30,83)(31,88)(32,57)(33,90)(34,59)(35,92)(36,61)(37,94)(38,63)(39,96)(40,65)(41,98)(42,67)(43,100)(44,69)(45,102)(46,71)(47,104)(48,73)(49,106)(50,75)(51,108)(52,77)(53,110)(54,79)(55,112)(56,81), (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), (1,101)(2,100)(3,99)(4,98)(5,97)(6,96)(7,95)(8,94)(9,93)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,57)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)>;
G:=Group( (1,84,45,88)(2,89,46,57)(3,58,47,90)(4,91,48,59)(5,60,49,92)(6,93,50,61)(7,62,51,94)(8,95,52,63)(9,64,53,96)(10,97,54,65)(11,66,55,98)(12,99,56,67)(13,68,29,100)(14,101,30,69)(15,70,31,102)(16,103,32,71)(17,72,33,104)(18,105,34,73)(19,74,35,106)(20,107,36,75)(21,76,37,108)(22,109,38,77)(23,78,39,110)(24,111,40,79)(25,80,41,112)(26,85,42,81)(27,82,43,86)(28,87,44,83), (1,70)(2,103)(3,72)(4,105)(5,74)(6,107)(7,76)(8,109)(9,78)(10,111)(11,80)(12,85)(13,82)(14,87)(15,84)(16,89)(17,58)(18,91)(19,60)(20,93)(21,62)(22,95)(23,64)(24,97)(25,66)(26,99)(27,68)(28,101)(29,86)(30,83)(31,88)(32,57)(33,90)(34,59)(35,92)(36,61)(37,94)(38,63)(39,96)(40,65)(41,98)(42,67)(43,100)(44,69)(45,102)(46,71)(47,104)(48,73)(49,106)(50,75)(51,108)(52,77)(53,110)(54,79)(55,112)(56,81), (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), (1,101)(2,100)(3,99)(4,98)(5,97)(6,96)(7,95)(8,94)(9,93)(10,92)(11,91)(12,90)(13,89)(14,88)(15,87)(16,86)(17,85)(18,112)(19,111)(20,110)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,57)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58) );
G=PermutationGroup([(1,84,45,88),(2,89,46,57),(3,58,47,90),(4,91,48,59),(5,60,49,92),(6,93,50,61),(7,62,51,94),(8,95,52,63),(9,64,53,96),(10,97,54,65),(11,66,55,98),(12,99,56,67),(13,68,29,100),(14,101,30,69),(15,70,31,102),(16,103,32,71),(17,72,33,104),(18,105,34,73),(19,74,35,106),(20,107,36,75),(21,76,37,108),(22,109,38,77),(23,78,39,110),(24,111,40,79),(25,80,41,112),(26,85,42,81),(27,82,43,86),(28,87,44,83)], [(1,70),(2,103),(3,72),(4,105),(5,74),(6,107),(7,76),(8,109),(9,78),(10,111),(11,80),(12,85),(13,82),(14,87),(15,84),(16,89),(17,58),(18,91),(19,60),(20,93),(21,62),(22,95),(23,64),(24,97),(25,66),(26,99),(27,68),(28,101),(29,86),(30,83),(31,88),(32,57),(33,90),(34,59),(35,92),(36,61),(37,94),(38,63),(39,96),(40,65),(41,98),(42,67),(43,100),(44,69),(45,102),(46,71),(47,104),(48,73),(49,106),(50,75),(51,108),(52,77),(53,110),(54,79),(55,112),(56,81)], [(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)], [(1,101),(2,100),(3,99),(4,98),(5,97),(6,96),(7,95),(8,94),(9,93),(10,92),(11,91),(12,90),(13,89),(14,88),(15,87),(16,86),(17,85),(18,112),(19,111),(20,110),(21,109),(22,108),(23,107),(24,106),(25,105),(26,104),(27,103),(28,102),(29,57),(30,84),(31,83),(32,82),(33,81),(34,80),(35,79),(36,78),(37,77),(38,76),(39,75),(40,74),(41,73),(42,72),(43,71),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58)])
Matrix representation ►G ⊆ 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] >;
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 | C28⋊1D4 | 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 |
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