metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.31D4, C23.36D28, (C2×C8)⋊15D14, C22⋊C8⋊8D7, (C2×C14)⋊1SD16, C4.119(D4×D7), (C2×C28).42D4, (C2×C4).31D28, C2.D56⋊8C2, (C2×C56)⋊14C22, C14.7C22≀C2, C28.331(C2×D4), C7⋊1(C22⋊SD16), C4⋊Dic7⋊1C22, C14.7(C2×SD16), C28.48D4⋊1C2, C14.8(C8⋊C22), C22⋊3(C56⋊C2), (C22×D28).3C2, (C22×C4).81D14, (C22×C14).51D4, C2.11(C8⋊D14), (C2×C28).741C23, (C2×Dic14)⋊1C22, C22.104(C2×D28), C2.10(C22⋊D28), (C2×D28).191C22, (C22×C28).50C22, (C2×C56⋊C2)⋊9C2, (C7×C22⋊C8)⋊10C2, C2.10(C2×C56⋊C2), (C2×C14).124(C2×D4), (C2×C4).686(C22×D7), SmallGroup(448,265)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.31D4
G = < a,b,c,d | a28=b2=d2=1, c4=a14, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=a7c3 >
Subgroups: 1404 in 188 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C56, Dic14, D28, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×C14, C22⋊SD16, C56⋊C2, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×C56, C2×Dic14, C2×D28, C2×D28, C22×C28, C23×D7, C2.D56, C7×C22⋊C8, C2×C56⋊C2, C28.48D4, C22×D28, D28.31D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8⋊C22, D28, C22×D7, C22⋊SD16, C56⋊C2, C2×D28, D4×D7, C22⋊D28, C2×C56⋊C2, C8⋊D14, D28.31D4
►On 112 points
Generators in S112
(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 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 112)(76 111)(77 110)(78 109)(79 108)(80 107)(81 106)(82 105)(83 104)(84 103)
(1 104 35 84 15 90 49 70)(2 105 36 57 16 91 50 71)(3 106 37 58 17 92 51 72)(4 107 38 59 18 93 52 73)(5 108 39 60 19 94 53 74)(6 109 40 61 20 95 54 75)(7 110 41 62 21 96 55 76)(8 111 42 63 22 97 56 77)(9 112 43 64 23 98 29 78)(10 85 44 65 24 99 30 79)(11 86 45 66 25 100 31 80)(12 87 46 67 26 101 32 81)(13 88 47 68 27 102 33 82)(14 89 48 69 28 103 34 83)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 112)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)
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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103), (1,104,35,84,15,90,49,70)(2,105,36,57,16,91,50,71)(3,106,37,58,17,92,51,72)(4,107,38,59,18,93,52,73)(5,108,39,60,19,94,53,74)(6,109,40,61,20,95,54,75)(7,110,41,62,21,96,55,76)(8,111,42,63,22,97,56,77)(9,112,43,64,23,98,29,78)(10,85,44,65,24,99,30,79)(11,86,45,66,25,100,31,80)(12,87,46,67,26,101,32,81)(13,88,47,68,27,102,33,82)(14,89,48,69,28,103,34,83), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,112)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111)>;
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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103), (1,104,35,84,15,90,49,70)(2,105,36,57,16,91,50,71)(3,106,37,58,17,92,51,72)(4,107,38,59,18,93,52,73)(5,108,39,60,19,94,53,74)(6,109,40,61,20,95,54,75)(7,110,41,62,21,96,55,76)(8,111,42,63,22,97,56,77)(9,112,43,64,23,98,29,78)(10,85,44,65,24,99,30,79)(11,86,45,66,25,100,31,80)(12,87,46,67,26,101,32,81)(13,88,47,68,27,102,33,82)(14,89,48,69,28,103,34,83), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,112)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111) );
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)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(57,102),(58,101),(59,100),(60,99),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,112),(76,111),(77,110),(78,109),(79,108),(80,107),(81,106),(82,105),(83,104),(84,103)], [(1,104,35,84,15,90,49,70),(2,105,36,57,16,91,50,71),(3,106,37,58,17,92,51,72),(4,107,38,59,18,93,52,73),(5,108,39,60,19,94,53,74),(6,109,40,61,20,95,54,75),(7,110,41,62,21,96,55,76),(8,111,42,63,22,97,56,77),(9,112,43,64,23,98,29,78),(10,85,44,65,24,99,30,79),(11,86,45,66,25,100,31,80),(12,87,46,67,26,101,32,81),(13,88,47,68,27,102,33,82),(14,89,48,69,28,103,34,83)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,112),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108),(82,109),(83,110),(84,111)]])
79 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 | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| 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 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | SD16 | D14 | D14 | D28 | D28 | C56⋊C2 | C8⋊C22 | D4×D7 | C8⋊D14 |
| kernel | D28.31D4 | C2.D56 | C7×C22⋊C8 | C2×C56⋊C2 | C28.48D4 | C22×D28 | D28 | C2×C28 | C22×C14 | C22⋊C8 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C14 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 3 | 4 | 6 | 3 | 6 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of D28.31D4 ►in GL4(𝔽113) generated by
| 109 | 32 | 0 | 0 |
| 81 | 58 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 109 | 32 | 0 | 0 |
| 109 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 112 |
| 89 | 71 | 0 | 0 |
| 42 | 50 | 0 | 0 |
| 0 | 0 | 112 | 2 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 112 |
G:=sub<GL(4,GF(113))| [109,81,0,0,32,58,0,0,0,0,112,0,0,0,0,112],[109,109,0,0,32,4,0,0,0,0,1,1,0,0,0,112],[89,42,0,0,71,50,0,0,0,0,112,0,0,0,2,1],[112,0,0,0,0,112,0,0,0,0,1,1,0,0,0,112] >;
D28.31D4 in GAP, Magma, Sage, TeX
D_{28}._{31}D_4 % in TeX
G:=Group("D28.31D4"); // GroupNames label
G:=SmallGroup(448,265);
// by ID
G=gap.SmallGroup(448,265);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=d^2=1,c^4=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=a^7*c^3>;
// generators/relations