metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.39D4, M4(2)⋊6D14, Dic14.39D4, (C2×Q8)⋊4D14, C4○D4.7D14, (C7×D4).14D4, C4.106(D4×D7), (C7×Q8).14D4, C8.C22⋊1D7, C28.198(C2×D4), C7⋊4(D4.9D4), D28⋊4C4⋊11C2, (Q8×C14)⋊4C22, (C22×D7).6D4, C22.37(D4×D7), C14.65C22≀C2, D4⋊2Dic7⋊8C2, D4⋊8D14.2C2, C28.C23⋊5C2, C28.23D4⋊7C2, D4.11(C7⋊D4), (C2×C28).17C23, Q8.11(C7⋊D4), (C4×Dic7)⋊6C22, C4.Dic7⋊9C22, C28.46D4⋊11C2, C4○D28.25C22, C2.33(C23⋊D14), (C2×D28).130C22, (C7×M4(2))⋊16C22, C4.54(C2×C7⋊D4), (C2×C14).36(C2×D4), (C7×C8.C22)⋊5C2, (C2×C4).17(C22×D7), (C7×C4○D4).15C22, SmallGroup(448,736)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.39D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=dad=a-1, cac-1=a13, cbc-1=a5b, dbd=a26b, dcd=a14c-1 >
Subgroups: 972 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C4.4D4, C8.C22, C8.C22, 2+ 1+4, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×D7, D4.9D4, C4.Dic7, C4×Dic7, D14⋊C4, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, Q8⋊2D7, Q8×C14, C7×C4○D4, C28.46D4, D28⋊4C4, D4⋊2Dic7, C28.C23, C28.23D4, C7×C8.C22, D4⋊8D14, D28.39D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4.9D4, D4×D7, C2×C7⋊D4, C23⋊D14, D28.39D4
►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 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 112)(12 111)(13 110)(14 109)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 98)(26 97)(27 96)(28 95)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 84)
(2 14)(3 27)(4 12)(5 25)(6 10)(7 23)(9 21)(11 19)(13 17)(16 28)(18 26)(20 24)(29 51)(30 36)(31 49)(32 34)(33 47)(35 45)(37 43)(38 56)(39 41)(40 54)(42 52)(44 50)(46 48)(53 55)(57 68 71 82)(58 81 72 67)(59 66 73 80)(60 79 74 65)(61 64 75 78)(62 77 76 63)(69 84 83 70)(85 98 99 112)(86 111 100 97)(87 96 101 110)(88 109 102 95)(89 94 103 108)(90 107 104 93)(91 92 105 106)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(57 104)(58 103)(59 102)(60 101)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 105)
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), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84), (2,14)(3,27)(4,12)(5,25)(6,10)(7,23)(9,21)(11,19)(13,17)(16,28)(18,26)(20,24)(29,51)(30,36)(31,49)(32,34)(33,47)(35,45)(37,43)(38,56)(39,41)(40,54)(42,52)(44,50)(46,48)(53,55)(57,68,71,82)(58,81,72,67)(59,66,73,80)(60,79,74,65)(61,64,75,78)(62,77,76,63)(69,84,83,70)(85,98,99,112)(86,111,100,97)(87,96,101,110)(88,109,102,95)(89,94,103,108)(90,107,104,93)(91,92,105,106), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)>;
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), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,106)(18,105)(19,104)(20,103)(21,102)(22,101)(23,100)(24,99)(25,98)(26,97)(27,96)(28,95)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84), (2,14)(3,27)(4,12)(5,25)(6,10)(7,23)(9,21)(11,19)(13,17)(16,28)(18,26)(20,24)(29,51)(30,36)(31,49)(32,34)(33,47)(35,45)(37,43)(38,56)(39,41)(40,54)(42,52)(44,50)(46,48)(53,55)(57,68,71,82)(58,81,72,67)(59,66,73,80)(60,79,74,65)(61,64,75,78)(62,77,76,63)(69,84,83,70)(85,98,99,112)(86,111,100,97)(87,96,101,110)(88,109,102,95)(89,94,103,108)(90,107,104,93)(91,92,105,106), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105) );
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)], [(1,94),(2,93),(3,92),(4,91),(5,90),(6,89),(7,88),(8,87),(9,86),(10,85),(11,112),(12,111),(13,110),(14,109),(15,108),(16,107),(17,106),(18,105),(19,104),(20,103),(21,102),(22,101),(23,100),(24,99),(25,98),(26,97),(27,96),(28,95),(29,83),(30,82),(31,81),(32,80),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(56,84)], [(2,14),(3,27),(4,12),(5,25),(6,10),(7,23),(9,21),(11,19),(13,17),(16,28),(18,26),(20,24),(29,51),(30,36),(31,49),(32,34),(33,47),(35,45),(37,43),(38,56),(39,41),(40,54),(42,52),(44,50),(46,48),(53,55),(57,68,71,82),(58,81,72,67),(59,66,73,80),(60,79,74,65),(61,64,75,78),(62,77,76,63),(69,84,83,70),(85,98,99,112),(86,111,100,97),(87,96,101,110),(88,109,102,95),(89,94,103,108),(90,107,104,93),(91,92,105,106)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(57,104),(58,103),(59,102),(60,101),(61,100),(62,99),(63,98),(64,97),(65,96),(66,95),(67,94),(68,93),(69,92),(70,91),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,112),(78,111),(79,110),(80,109),(81,108),(82,107),(83,106),(84,105)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 28 | 28 | 2 | 2 | 4 | 8 | 28 | 28 | 28 | 2 | 2 | 2 | 8 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | D4.9D4 | D4×D7 | D4×D7 | D28.39D4 |
| kernel | D28.39D4 | C28.46D4 | D28⋊4C4 | D4⋊2Dic7 | C28.C23 | C28.23D4 | C7×C8.C22 | D4⋊8D14 | Dic14 | D28 | C7×D4 | C7×Q8 | C22×D7 | C8.C22 | M4(2) | C2×Q8 | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 3 |
Matrix representation of D28.39D4 ►in GL8(𝔽113)
| 55 | 79 | 0 | 0 | 0 | 0 | 0 | 0 |
| 68 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 68 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 15 | 15 |
| 5 | 48 | 42 | 19 | 0 | 0 | 0 | 0 |
| 16 | 108 | 44 | 71 | 0 | 0 | 0 | 0 |
| 71 | 94 | 108 | 65 | 0 | 0 | 0 | 0 |
| 69 | 42 | 97 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 | 57 | 98 | 83 |
| 0 | 0 | 0 | 0 | 54 | 63 | 62 | 62 |
| 0 | 0 | 0 | 0 | 98 | 62 | 0 | 0 |
| 0 | 0 | 0 | 0 | 104 | 54 | 74 | 98 |
| 58 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 44 | 55 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 4 | 60 | 0 | 15 |
| 0 | 0 | 58 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 55 | 0 | 0 | 0 | 0 |
| 58 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 44 | 55 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 87 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 49 | 112 | 111 |
| 0 | 0 | 0 | 0 | 71 | 19 | 0 | 1 |
G:=sub<GL(8,GF(113))| [55,68,0,0,0,0,0,0,79,34,0,0,0,0,0,0,0,0,55,68,0,0,0,0,0,0,79,34,0,0,0,0,0,0,0,0,98,62,0,98,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98,15,0,0,0,0,0,0,0,15],[5,16,71,69,0,0,0,0,48,108,94,42,0,0,0,0,42,44,108,97,0,0,0,0,19,71,65,5,0,0,0,0,0,0,0,0,65,54,98,104,0,0,0,0,57,63,62,54,0,0,0,0,98,62,0,74,0,0,0,0,83,62,0,98],[58,44,0,0,0,0,0,0,34,55,0,0,0,0,0,0,0,0,58,44,0,0,0,0,0,0,34,55,0,0,0,0,0,0,0,0,1,26,0,4,0,0,0,0,0,112,0,60,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,15],[0,0,58,44,0,0,0,0,0,0,34,55,0,0,0,0,58,44,0,0,0,0,0,0,34,55,0,0,0,0,0,0,0,0,0,0,112,0,42,71,0,0,0,0,87,1,49,19,0,0,0,0,0,0,112,0,0,0,0,0,0,0,111,1] >;
D28.39D4 in GAP, Magma, Sage, TeX
D_{28}._{39}D_4 % in TeX
G:=Group("D28.39D4"); // GroupNames label
G:=SmallGroup(448,736);
// by ID
G=gap.SmallGroup(448,736);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,184,570,1684,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^5*b,d*b*d=a^26*b,d*c*d=a^14*c^-1>;
// generators/relations