metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.5D4, Dic14.5D4, M4(2).4D14, (C2×C4).7D28, (C2×C28).9D4, C4.81(D4×D7), C8⋊D14⋊7C2, C28.98(C2×D4), (C2×Q8).6D14, D28⋊4C4⋊4C2, C4.10D4⋊2D7, C7⋊1(D4.8D4), C14.18C22≀C2, C28.23D4⋊1C2, (C2×C28).10C23, C4○D28.6C22, C22.13(C2×D28), (Q8×C14).8C22, Q8.10D14⋊1C2, (C2×D28).41C22, C2.21(C22⋊D28), (C4×Dic7).2C22, (C7×M4(2)).3C22, (C2×C14).23(C2×D4), (C7×C4.10D4)⋊4C2, (C2×C4).10(C22×D7), SmallGroup(448,287)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.5D4
G = < a,b,c,d | a28=b2=1, c4=a14, d2=a21, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, dbd-1=a21b, dcd-1=a21c3 >
Subgroups: 940 in 146 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, D4.8D4, C56⋊C2, D56, C4×Dic7, D14⋊C4, C7×M4(2), C2×D28, C4○D28, C4○D28, Q8×D7, Q8⋊2D7, Q8×C14, D28⋊4C4, C7×C4.10D4, C8⋊D14, C28.23D4, Q8.10D14, D28.5D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.8D4, C2×D28, D4×D7, C22⋊D28, D28.5D4
►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 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(57 86)(58 85)(59 112)(60 111)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 90)(82 89)(83 88)(84 87)
(1 43 22 50 15 29 8 36)(2 30 23 37 16 44 9 51)(3 45 24 52 17 31 10 38)(4 32 25 39 18 46 11 53)(5 47 26 54 19 33 12 40)(6 34 27 41 20 48 13 55)(7 49 28 56 21 35 14 42)(57 90 78 97 71 104 64 111)(58 105 79 112 72 91 65 98)(59 92 80 99 73 106 66 85)(60 107 81 86 74 93 67 100)(61 94 82 101 75 108 68 87)(62 109 83 88 76 95 69 102)(63 96 84 103 77 110 70 89)
(1 85 22 106 15 99 8 92)(2 86 23 107 16 100 9 93)(3 87 24 108 17 101 10 94)(4 88 25 109 18 102 11 95)(5 89 26 110 19 103 12 96)(6 90 27 111 20 104 13 97)(7 91 28 112 21 105 14 98)(29 59 50 80 43 73 36 66)(30 60 51 81 44 74 37 67)(31 61 52 82 45 75 38 68)(32 62 53 83 46 76 39 69)(33 63 54 84 47 77 40 70)(34 64 55 57 48 78 41 71)(35 65 56 58 49 79 42 72)
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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(57,86)(58,85)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87), (1,43,22,50,15,29,8,36)(2,30,23,37,16,44,9,51)(3,45,24,52,17,31,10,38)(4,32,25,39,18,46,11,53)(5,47,26,54,19,33,12,40)(6,34,27,41,20,48,13,55)(7,49,28,56,21,35,14,42)(57,90,78,97,71,104,64,111)(58,105,79,112,72,91,65,98)(59,92,80,99,73,106,66,85)(60,107,81,86,74,93,67,100)(61,94,82,101,75,108,68,87)(62,109,83,88,76,95,69,102)(63,96,84,103,77,110,70,89), (1,85,22,106,15,99,8,92)(2,86,23,107,16,100,9,93)(3,87,24,108,17,101,10,94)(4,88,25,109,18,102,11,95)(5,89,26,110,19,103,12,96)(6,90,27,111,20,104,13,97)(7,91,28,112,21,105,14,98)(29,59,50,80,43,73,36,66)(30,60,51,81,44,74,37,67)(31,61,52,82,45,75,38,68)(32,62,53,83,46,76,39,69)(33,63,54,84,47,77,40,70)(34,64,55,57,48,78,41,71)(35,65,56,58,49,79,42,72)>;
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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(57,86)(58,85)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87), (1,43,22,50,15,29,8,36)(2,30,23,37,16,44,9,51)(3,45,24,52,17,31,10,38)(4,32,25,39,18,46,11,53)(5,47,26,54,19,33,12,40)(6,34,27,41,20,48,13,55)(7,49,28,56,21,35,14,42)(57,90,78,97,71,104,64,111)(58,105,79,112,72,91,65,98)(59,92,80,99,73,106,66,85)(60,107,81,86,74,93,67,100)(61,94,82,101,75,108,68,87)(62,109,83,88,76,95,69,102)(63,96,84,103,77,110,70,89), (1,85,22,106,15,99,8,92)(2,86,23,107,16,100,9,93)(3,87,24,108,17,101,10,94)(4,88,25,109,18,102,11,95)(5,89,26,110,19,103,12,96)(6,90,27,111,20,104,13,97)(7,91,28,112,21,105,14,98)(29,59,50,80,43,73,36,66)(30,60,51,81,44,74,37,67)(31,61,52,82,45,75,38,68)(32,62,53,83,46,76,39,69)(33,63,54,84,47,77,40,70)(34,64,55,57,48,78,41,71)(35,65,56,58,49,79,42,72) );
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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(57,86),(58,85),(59,112),(60,111),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,90),(82,89),(83,88),(84,87)], [(1,43,22,50,15,29,8,36),(2,30,23,37,16,44,9,51),(3,45,24,52,17,31,10,38),(4,32,25,39,18,46,11,53),(5,47,26,54,19,33,12,40),(6,34,27,41,20,48,13,55),(7,49,28,56,21,35,14,42),(57,90,78,97,71,104,64,111),(58,105,79,112,72,91,65,98),(59,92,80,99,73,106,66,85),(60,107,81,86,74,93,67,100),(61,94,82,101,75,108,68,87),(62,109,83,88,76,95,69,102),(63,96,84,103,77,110,70,89)], [(1,85,22,106,15,99,8,92),(2,86,23,107,16,100,9,93),(3,87,24,108,17,101,10,94),(4,88,25,109,18,102,11,95),(5,89,26,110,19,103,12,96),(6,90,27,111,20,104,13,97),(7,91,28,112,21,105,14,98),(29,59,50,80,43,73,36,66),(30,60,51,81,44,74,37,67),(31,61,52,82,45,75,38,68),(32,62,53,83,46,76,39,69),(33,63,54,84,47,77,40,70),(34,64,55,57,48,78,41,71),(35,65,56,58,49,79,42,72)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 28 | 56 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D28 | D4.8D4 | D4×D7 | D28.5D4 |
| kernel | D28.5D4 | D28⋊4C4 | C7×C4.10D4 | C8⋊D14 | C28.23D4 | Q8.10D14 | Dic14 | D28 | C2×C28 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | C7 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 6 | 3 | 12 | 2 | 6 | 3 |
Matrix representation of D28.5D4 ►in GL8(𝔽113)
| 34 | 0 | 89 | 0 | 0 | 0 | 0 | 0 |
| 0 | 34 | 0 | 89 | 0 | 0 | 0 | 0 |
| 59 | 0 | 88 | 0 | 0 | 0 | 0 | 0 |
| 0 | 59 | 0 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 | 0 | 0 |
| 34 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 98 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 0 | 0 |
G:=sub<GL(8,GF(113))| [34,0,59,0,0,0,0,0,0,34,0,59,0,0,0,0,89,0,88,0,0,0,0,0,0,89,0,88,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,15],[0,0,34,0,0,0,0,0,0,0,0,34,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,98,0],[0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,112,0],[112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,0,0,0,0,0,112,0,0,0] >;
D28.5D4 in GAP, Magma, Sage, TeX
D_{28}._5D_4 % in TeX
G:=Group("D28.5D4"); // GroupNames label
G:=SmallGroup(448,287);
// by ID
G=gap.SmallGroup(448,287);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,226,1123,570,136,1684,438,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=a^14,d^2=a^21,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^21*c^3>;
// generators/relations