metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.3D28, C56.87D4, Q8.3D28, M4(2).33D14, C8○D4⋊1D7, (C7×D4).20D4, (C2×C8).80D14, C28.42(C2×D4), C4.19(C2×D28), (C7×Q8).20D4, C4○D4.31D14, C7⋊4(D4.3D4), C8.44(C7⋊D4), C56.C4⋊14C2, D4⋊D14.1C2, D4.9D14⋊3C2, (C2×C56).66C22, C28.46D4⋊13C2, C4.12D28⋊13C2, C2.24(C28⋊7D4), C14.76(C4⋊D4), (C2×C28).421C23, C22.8(C4○D28), (C2×D28).111C22, C4.Dic7.16C22, (C7×M4(2)).36C22, (C2×Dic14).117C22, (C7×C8○D4)⋊1C2, (C2×C56⋊C2)⋊3C2, C4.117(C2×C7⋊D4), (C2×C14).6(C4○D4), (C7×C4○D4).36C22, (C2×C4).123(C22×D7), SmallGroup(448,675)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.3D28
G = < a,b,c,d | a4=b2=1, c28=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c27 >
Subgroups: 572 in 104 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C56, C56, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, D4.3D4, C56⋊C2, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C2×Dic14, C2×D28, C7×C4○D4, C56.C4, C28.46D4, C4.12D28, C2×C56⋊C2, D4⋊D14, D4.9D14, C7×C8○D4, D4.3D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, D28, C7⋊D4, C22×D7, D4.3D4, C2×D28, C4○D28, C2×C7⋊D4, C28⋊7D4, D4.3D28
►On 112 points
Generators in S112
(1 15 29 43)(2 16 30 44)(3 17 31 45)(4 18 32 46)(5 19 33 47)(6 20 34 48)(7 21 35 49)(8 22 36 50)(9 23 37 51)(10 24 38 52)(11 25 39 53)(12 26 40 54)(13 27 41 55)(14 28 42 56)(57 99 85 71)(58 100 86 72)(59 101 87 73)(60 102 88 74)(61 103 89 75)(62 104 90 76)(63 105 91 77)(64 106 92 78)(65 107 93 79)(66 108 94 80)(67 109 95 81)(68 110 96 82)(69 111 97 83)(70 112 98 84)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)
(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 28 29 56)(2 55 30 27)(3 26 31 54)(4 53 32 25)(5 24 33 52)(6 51 34 23)(7 22 35 50)(8 49 36 21)(9 20 37 48)(10 47 38 19)(11 18 39 46)(12 45 40 17)(13 16 41 44)(14 43 42 15)(57 92 85 64)(58 63 86 91)(59 90 87 62)(60 61 88 89)(65 84 93 112)(66 111 94 83)(67 82 95 110)(68 109 96 81)(69 80 97 108)(70 107 98 79)(71 78 99 106)(72 105 100 77)(73 76 101 104)(74 103 102 75)
G:=sub<Sym(112)| (1,15,29,43)(2,16,30,44)(3,17,31,45)(4,18,32,46)(5,19,33,47)(6,20,34,48)(7,21,35,49)(8,22,36,50)(9,23,37,51)(10,24,38,52)(11,25,39,53)(12,26,40,54)(13,27,41,55)(14,28,42,56)(57,99,85,71)(58,100,86,72)(59,101,87,73)(60,102,88,74)(61,103,89,75)(62,104,90,76)(63,105,91,77)(64,106,92,78)(65,107,93,79)(66,108,94,80)(67,109,95,81)(68,110,96,82)(69,111,97,83)(70,112,98,84), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95), (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), (1,28,29,56)(2,55,30,27)(3,26,31,54)(4,53,32,25)(5,24,33,52)(6,51,34,23)(7,22,35,50)(8,49,36,21)(9,20,37,48)(10,47,38,19)(11,18,39,46)(12,45,40,17)(13,16,41,44)(14,43,42,15)(57,92,85,64)(58,63,86,91)(59,90,87,62)(60,61,88,89)(65,84,93,112)(66,111,94,83)(67,82,95,110)(68,109,96,81)(69,80,97,108)(70,107,98,79)(71,78,99,106)(72,105,100,77)(73,76,101,104)(74,103,102,75)>;
G:=Group( (1,15,29,43)(2,16,30,44)(3,17,31,45)(4,18,32,46)(5,19,33,47)(6,20,34,48)(7,21,35,49)(8,22,36,50)(9,23,37,51)(10,24,38,52)(11,25,39,53)(12,26,40,54)(13,27,41,55)(14,28,42,56)(57,99,85,71)(58,100,86,72)(59,101,87,73)(60,102,88,74)(61,103,89,75)(62,104,90,76)(63,105,91,77)(64,106,92,78)(65,107,93,79)(66,108,94,80)(67,109,95,81)(68,110,96,82)(69,111,97,83)(70,112,98,84), (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,57)(19,58)(20,59)(21,60)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95), (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), (1,28,29,56)(2,55,30,27)(3,26,31,54)(4,53,32,25)(5,24,33,52)(6,51,34,23)(7,22,35,50)(8,49,36,21)(9,20,37,48)(10,47,38,19)(11,18,39,46)(12,45,40,17)(13,16,41,44)(14,43,42,15)(57,92,85,64)(58,63,86,91)(59,90,87,62)(60,61,88,89)(65,84,93,112)(66,111,94,83)(67,82,95,110)(68,109,96,81)(69,80,97,108)(70,107,98,79)(71,78,99,106)(72,105,100,77)(73,76,101,104)(74,103,102,75) );
G=PermutationGroup([[(1,15,29,43),(2,16,30,44),(3,17,31,45),(4,18,32,46),(5,19,33,47),(6,20,34,48),(7,21,35,49),(8,22,36,50),(9,23,37,51),(10,24,38,52),(11,25,39,53),(12,26,40,54),(13,27,41,55),(14,28,42,56),(57,99,85,71),(58,100,86,72),(59,101,87,73),(60,102,88,74),(61,103,89,75),(62,104,90,76),(63,105,91,77),(64,106,92,78),(65,107,93,79),(66,108,94,80),(67,109,95,81),(68,110,96,82),(69,111,97,83),(70,112,98,84)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,57),(19,58),(20,59),(21,60),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95)], [(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)], [(1,28,29,56),(2,55,30,27),(3,26,31,54),(4,53,32,25),(5,24,33,52),(6,51,34,23),(7,22,35,50),(8,49,36,21),(9,20,37,48),(10,47,38,19),(11,18,39,46),(12,45,40,17),(13,16,41,44),(14,43,42,15),(57,92,85,64),(58,63,86,91),(59,90,87,62),(60,61,88,89),(65,84,93,112),(66,111,94,83),(67,82,95,110),(68,109,96,81),(69,80,97,108),(70,107,98,79),(71,78,99,106),(72,105,100,77),(73,76,101,104),(74,103,102,75)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56L | 56M | ··· | 56AD |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 56 | 2 | 2 | 4 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | D28 | D28 | C4○D28 | D4.3D4 | D4.3D28 |
| kernel | D4.3D28 | C56.C4 | C28.46D4 | C4.12D28 | C2×C56⋊C2 | D4⋊D14 | D4.9D14 | C7×C8○D4 | C56 | C7×D4 | C7×Q8 | C8○D4 | C2×C14 | C2×C8 | M4(2) | C4○D4 | C8 | D4 | Q8 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 12 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of D4.3D28 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 97 | 0 | 0 | 0 | 0 | 0 |
| 90 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 100 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 97 | 16 | 0 | 0 | 0 | 0 |
| 90 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 100 | 0 | 0 |
| 0 | 0 | 100 | 100 | 0 | 0 |
| 0 | 0 | 0 | 0 | 100 | 100 |
| 0 | 0 | 0 | 0 | 100 | 13 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[1,2,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,1,0,0,0],[97,90,0,0,0,0,0,7,0,0,0,0,0,0,13,13,0,0,0,0,100,13,0,0,0,0,0,0,13,13,0,0,0,0,100,13],[97,90,0,0,0,0,16,16,0,0,0,0,0,0,13,100,0,0,0,0,100,100,0,0,0,0,0,0,100,100,0,0,0,0,100,13] >;
D4.3D28 in GAP, Magma, Sage, TeX
D_4._3D_{28} % in TeX
G:=Group("D4.3D28"); // GroupNames label
G:=SmallGroup(448,675);
// by ID
G=gap.SmallGroup(448,675);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,1123,297,136,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^28=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^27>;
// generators/relations