metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.36D4, C4⋊C4⋊4D14, (C2×Q8)⋊1D14, C22⋊Q8⋊1D7, (C2×C14)⋊5SD16, (C2×C28).77D4, C4.101(D4×D7), C28.152(C2×D4), C7⋊3(C22⋊SD16), C22⋊2(Q8⋊D7), C14.D8⋊37C2, (Q8×C14)⋊1C22, C14.47C22≀C2, C14.72(C2×SD16), (C22×C14).92D4, C28.55D4⋊13C2, (C2×C28).365C23, (C22×D28).13C2, (C22×C4).127D14, C23.62(C7⋊D4), C2.15(D4⋊D14), C2.15(C23⋊D14), C14.116(C8⋊C22), (C2×D28).242C22, (C22×C28).169C22, (C2×C7⋊C8)⋊6C22, (C2×Q8⋊D7)⋊8C2, C2.9(C2×Q8⋊D7), (C7×C4⋊C4)⋊6C22, (C7×C22⋊Q8)⋊1C2, (C2×C14).496(C2×D4), (C2×C4).55(C7⋊D4), (C2×C4).465(C22×D7), C22.171(C2×C7⋊D4), SmallGroup(448,580)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.36D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, bd=db, dcd=a14c-1 >
Subgroups: 1308 in 188 conjugacy classes, 47 normal (27 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, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C7⋊C8, D28, D28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×C14, C22⋊SD16, C2×C7⋊C8, Q8⋊D7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×D28, C2×D28, C22×C28, Q8×C14, C23×D7, C14.D8, C28.55D4, C2×Q8⋊D7, C7×C22⋊Q8, C22×D28, D28.36D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8⋊C22, C7⋊D4, C22×D7, C22⋊SD16, Q8⋊D7, D4×D7, C2×C7⋊D4, C23⋊D14, C2×Q8⋊D7, D4⋊D14, D28.36D4
►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 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 99)(30 98)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)
(1 30 77 92)(2 45 78 107)(3 32 79 94)(4 47 80 109)(5 34 81 96)(6 49 82 111)(7 36 83 98)(8 51 84 85)(9 38 57 100)(10 53 58 87)(11 40 59 102)(12 55 60 89)(13 42 61 104)(14 29 62 91)(15 44 63 106)(16 31 64 93)(17 46 65 108)(18 33 66 95)(19 48 67 110)(20 35 68 97)(21 50 69 112)(22 37 70 99)(23 52 71 86)(24 39 72 101)(25 54 73 88)(26 41 74 103)(27 56 75 90)(28 43 76 105)
(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 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)
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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,99)(30,98)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100), (1,30,77,92)(2,45,78,107)(3,32,79,94)(4,47,80,109)(5,34,81,96)(6,49,82,111)(7,36,83,98)(8,51,84,85)(9,38,57,100)(10,53,58,87)(11,40,59,102)(12,55,60,89)(13,42,61,104)(14,29,62,91)(15,44,63,106)(16,31,64,93)(17,46,65,108)(18,33,66,95)(19,48,67,110)(20,35,68,97)(21,50,69,112)(22,37,70,99)(23,52,71,86)(24,39,72,101)(25,54,73,88)(26,41,74,103)(27,56,75,90)(28,43,76,105), (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)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)>;
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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,99)(30,98)(31,97)(32,96)(33,95)(34,94)(35,93)(36,92)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,112)(45,111)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100), (1,30,77,92)(2,45,78,107)(3,32,79,94)(4,47,80,109)(5,34,81,96)(6,49,82,111)(7,36,83,98)(8,51,84,85)(9,38,57,100)(10,53,58,87)(11,40,59,102)(12,55,60,89)(13,42,61,104)(14,29,62,91)(15,44,63,106)(16,31,64,93)(17,46,65,108)(18,33,66,95)(19,48,67,110)(20,35,68,97)(21,50,69,112)(22,37,70,99)(23,52,71,86)(24,39,72,101)(25,54,73,88)(26,41,74,103)(27,56,75,90)(28,43,76,105), (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)(29,91)(30,92)(31,93)(32,94)(33,95)(34,96)(35,97)(36,98)(37,99)(38,100)(39,101)(40,102)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84) );
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)], [(1,62),(2,61),(3,60),(4,59),(5,58),(6,57),(7,84),(8,83),(9,82),(10,81),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,99),(30,98),(31,97),(32,96),(33,95),(34,94),(35,93),(36,92),(37,91),(38,90),(39,89),(40,88),(41,87),(42,86),(43,85),(44,112),(45,111),(46,110),(47,109),(48,108),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,100)], [(1,30,77,92),(2,45,78,107),(3,32,79,94),(4,47,80,109),(5,34,81,96),(6,49,82,111),(7,36,83,98),(8,51,84,85),(9,38,57,100),(10,53,58,87),(11,40,59,102),(12,55,60,89),(13,42,61,104),(14,29,62,91),(15,44,63,106),(16,31,64,93),(17,46,65,108),(18,33,66,95),(19,48,67,110),(20,35,68,97),(21,50,69,112),(22,37,70,99),(23,52,71,86),(24,39,72,101),(25,54,73,88),(26,41,74,103),(27,56,75,90),(28,43,76,105)], [(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),(29,91),(30,92),(31,93),(32,94),(33,95),(34,96),(35,97),(36,98),(37,99),(38,100),(39,101),(40,102),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84)]])
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 | ··· | 28L | 28M | ··· | 28X |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | SD16 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4×D7 | Q8⋊D7 | D4⋊D14 |
| kernel | D28.36D4 | C14.D8 | C28.55D4 | C2×Q8⋊D7 | C7×C22⋊Q8 | C22×D28 | D28 | C2×C28 | C22×C14 | C22⋊Q8 | C2×C14 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C23 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 6 | 6 | 1 | 6 | 6 | 6 |
Matrix representation of D28.36D4 ►in GL6(𝔽113)
| 0 | 104 | 0 | 0 | 0 | 0 |
| 88 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 77 | 41 | 0 | 0 |
| 0 | 0 | 29 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 80 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 77 | 41 | 0 | 0 |
| 0 | 0 | 7 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 110 | 81 | 0 | 0 |
| 0 | 0 | 78 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [0,88,0,0,0,0,104,79,0,0,0,0,0,0,77,29,0,0,0,0,41,36,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,80,1,0,0,0,0,0,0,77,7,0,0,0,0,41,36,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,110,78,0,0,0,0,81,3,0,0,0,0,0,0,0,1,0,0,0,0,112,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112] >;
D28.36D4 in GAP, Magma, Sage, TeX
D_{28}._{36}D_4 % in TeX
G:=Group("D28.36D4"); // GroupNames label
G:=SmallGroup(448,580);
// by ID
G=gap.SmallGroup(448,580);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,184,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=a^14*c^-1>;
// generators/relations