metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.8D4, D4.6D28, D14⋊5SD16, C4⋊C4⋊2D14, D14⋊C8⋊9C2, (C2×C8)⋊16D14, (C7×D4).1D4, C28.1(C2×D4), C4.2(C2×D28), C4.85(D4×D7), C14.D8⋊7C2, D4⋊C4⋊10D7, (C2×C56)⋊15C22, D14⋊2Q8⋊1C2, C7⋊2(C22⋊SD16), C2.11(D7×SD16), C14.20C22≀C2, (C2×D4).135D14, (C2×Dic7).20D4, C14.23(C2×SD16), (C22×D7).72D4, C22.174(D4×D7), C2.13(D8⋊D7), C14.31(C8⋊C22), (C2×C28).216C23, (D4×C14).37C22, (C2×D28).50C22, C2.23(C22⋊D28), (C2×Dic14)⋊13C22, (C2×D4×D7).5C2, (C2×C7⋊C8)⋊3C22, (C7×C4⋊C4)⋊4C22, (C2×D4.D7)⋊3C2, (C2×C56⋊C2)⋊14C2, (C2×C4×D7).9C22, (C7×D4⋊C4)⋊10C2, (C2×C14).229(C2×D4), (C2×C4).323(C22×D7), SmallGroup(448,310)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C22 — C2×C4 — D4⋊C4 |
Generators and relations for D28.8D4
G = < a,b,c,d | a28=b2=c4=1, d2=a7, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, dbd-1=a21b, dcd-1=a7c-1 >
Subgroups: 1332 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×20], C7, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×8], Q8 [×2], C23 [×11], D7 [×4], C14 [×3], C14 [×2], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×D4 [×6], C2×Q8, C24, Dic7 [×2], C28 [×2], C28, D14 [×2], D14 [×14], C2×C14, C2×C14 [×4], C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16 [×2], C22×D4, C7⋊C8, C56, Dic14 [×2], C4×D7 [×2], D28 [×2], D28, C2×Dic7, C2×Dic7, C7⋊D4 [×4], C2×C28, C2×C28, C7×D4 [×2], C7×D4, C22×D7, C22×D7 [×9], C22×C14, C22⋊SD16, C56⋊C2 [×2], C2×C7⋊C8, C4⋊Dic7, D14⋊C4, D4.D7 [×2], C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, D4×D7 [×4], C2×C7⋊D4, D4×C14, C23×D7, C14.D8, D14⋊C8, C7×D4⋊C4, D14⋊2Q8, C2×C56⋊C2, C2×D4.D7, C2×D4×D7, D28.8D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D7, SD16 [×2], C2×D4 [×3], D14 [×3], C22≀C2, C2×SD16, C8⋊C22, D28 [×2], C22×D7, C22⋊SD16, C2×D28, D4×D7 [×2], C22⋊D28, D8⋊D7, D7×SD16, D28.8D4
(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 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(49 56)(50 55)(51 54)(52 53)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)
(1 105 82 53)(2 92 83 40)(3 107 84 55)(4 94 57 42)(5 109 58 29)(6 96 59 44)(7 111 60 31)(8 98 61 46)(9 85 62 33)(10 100 63 48)(11 87 64 35)(12 102 65 50)(13 89 66 37)(14 104 67 52)(15 91 68 39)(16 106 69 54)(17 93 70 41)(18 108 71 56)(19 95 72 43)(20 110 73 30)(21 97 74 45)(22 112 75 32)(23 99 76 47)(24 86 77 34)(25 101 78 49)(26 88 79 36)(27 103 80 51)(28 90 81 38)
(1 53 8 32 15 39 22 46)(2 54 9 33 16 40 23 47)(3 55 10 34 17 41 24 48)(4 56 11 35 18 42 25 49)(5 29 12 36 19 43 26 50)(6 30 13 37 20 44 27 51)(7 31 14 38 21 45 28 52)(57 108 64 87 71 94 78 101)(58 109 65 88 72 95 79 102)(59 110 66 89 73 96 80 103)(60 111 67 90 74 97 81 104)(61 112 68 91 75 98 82 105)(62 85 69 92 76 99 83 106)(63 86 70 93 77 100 84 107)
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)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(49,56)(50,55)(51,54)(52,53)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105), (1,105,82,53)(2,92,83,40)(3,107,84,55)(4,94,57,42)(5,109,58,29)(6,96,59,44)(7,111,60,31)(8,98,61,46)(9,85,62,33)(10,100,63,48)(11,87,64,35)(12,102,65,50)(13,89,66,37)(14,104,67,52)(15,91,68,39)(16,106,69,54)(17,93,70,41)(18,108,71,56)(19,95,72,43)(20,110,73,30)(21,97,74,45)(22,112,75,32)(23,99,76,47)(24,86,77,34)(25,101,78,49)(26,88,79,36)(27,103,80,51)(28,90,81,38), (1,53,8,32,15,39,22,46)(2,54,9,33,16,40,23,47)(3,55,10,34,17,41,24,48)(4,56,11,35,18,42,25,49)(5,29,12,36,19,43,26,50)(6,30,13,37,20,44,27,51)(7,31,14,38,21,45,28,52)(57,108,64,87,71,94,78,101)(58,109,65,88,72,95,79,102)(59,110,66,89,73,96,80,103)(60,111,67,90,74,97,81,104)(61,112,68,91,75,98,82,105)(62,85,69,92,76,99,83,106)(63,86,70,93,77,100,84,107)>;
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)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(49,56)(50,55)(51,54)(52,53)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105), (1,105,82,53)(2,92,83,40)(3,107,84,55)(4,94,57,42)(5,109,58,29)(6,96,59,44)(7,111,60,31)(8,98,61,46)(9,85,62,33)(10,100,63,48)(11,87,64,35)(12,102,65,50)(13,89,66,37)(14,104,67,52)(15,91,68,39)(16,106,69,54)(17,93,70,41)(18,108,71,56)(19,95,72,43)(20,110,73,30)(21,97,74,45)(22,112,75,32)(23,99,76,47)(24,86,77,34)(25,101,78,49)(26,88,79,36)(27,103,80,51)(28,90,81,38), (1,53,8,32,15,39,22,46)(2,54,9,33,16,40,23,47)(3,55,10,34,17,41,24,48)(4,56,11,35,18,42,25,49)(5,29,12,36,19,43,26,50)(6,30,13,37,20,44,27,51)(7,31,14,38,21,45,28,52)(57,108,64,87,71,94,78,101)(58,109,65,88,72,95,79,102)(59,110,66,89,73,96,80,103)(60,111,67,90,74,97,81,104)(61,112,68,91,75,98,82,105)(62,85,69,92,76,99,83,106)(63,86,70,93,77,100,84,107) );
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),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(49,56),(50,55),(51,54),(52,53),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105)], [(1,105,82,53),(2,92,83,40),(3,107,84,55),(4,94,57,42),(5,109,58,29),(6,96,59,44),(7,111,60,31),(8,98,61,46),(9,85,62,33),(10,100,63,48),(11,87,64,35),(12,102,65,50),(13,89,66,37),(14,104,67,52),(15,91,68,39),(16,106,69,54),(17,93,70,41),(18,108,71,56),(19,95,72,43),(20,110,73,30),(21,97,74,45),(22,112,75,32),(23,99,76,47),(24,86,77,34),(25,101,78,49),(26,88,79,36),(27,103,80,51),(28,90,81,38)], [(1,53,8,32,15,39,22,46),(2,54,9,33,16,40,23,47),(3,55,10,34,17,41,24,48),(4,56,11,35,18,42,25,49),(5,29,12,36,19,43,26,50),(6,30,13,37,20,44,27,51),(7,31,14,38,21,45,28,52),(57,108,64,87,71,94,78,101),(58,109,65,88,72,95,79,102),(59,110,66,89,73,96,80,103),(60,111,67,90,74,97,81,104),(61,112,68,91,75,98,82,105),(62,85,69,92,76,99,83,106),(63,86,70,93,77,100,84,107)])
61 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 8 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | SD16 | D14 | D14 | D14 | D28 | C8⋊C22 | D4×D7 | D4×D7 | D8⋊D7 | D7×SD16 |
kernel | D28.8D4 | C14.D8 | D14⋊C8 | C7×D4⋊C4 | D14⋊2Q8 | C2×C56⋊C2 | C2×D4.D7 | C2×D4×D7 | D28 | C2×Dic7 | C7×D4 | C22×D7 | D4⋊C4 | D14 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C14 | C4 | C22 | C2 | C2 |
# reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 3 | 3 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D28.8D4 ►in GL4(𝔽113) generated by
112 | 72 | 0 | 0 |
91 | 1 | 0 | 0 |
0 | 0 | 89 | 79 |
0 | 0 | 10 | 0 |
1 | 0 | 0 | 0 |
22 | 112 | 0 | 0 |
0 | 0 | 0 | 79 |
0 | 0 | 103 | 0 |
26 | 81 | 0 | 0 |
60 | 87 | 0 | 0 |
0 | 0 | 17 | 67 |
0 | 0 | 80 | 96 |
0 | 81 | 0 | 0 |
60 | 87 | 0 | 0 |
0 | 0 | 96 | 46 |
0 | 0 | 33 | 17 |
G:=sub<GL(4,GF(113))| [112,91,0,0,72,1,0,0,0,0,89,10,0,0,79,0],[1,22,0,0,0,112,0,0,0,0,0,103,0,0,79,0],[26,60,0,0,81,87,0,0,0,0,17,80,0,0,67,96],[0,60,0,0,81,87,0,0,0,0,96,33,0,0,46,17] >;
D28.8D4 in GAP, Magma, Sage, TeX
D_{28}._8D_4
% in TeX
G:=Group("D28.8D4");
// GroupNames label
G:=SmallGroup(448,310);
// by ID
G=gap.SmallGroup(448,310);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,135,268,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^7,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,d*b*d^-1=a^21*b,d*c*d^-1=a^7*c^-1>;
// generators/relations