metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D28, C42⋊3D14, Q8.8D28, D28.33D4, M4(2)⋊4D14, Dic14.33D4, C4≀C2⋊2D7, (C7×D4).3D4, C28.4(C2×D4), (C7×Q8).3D4, C4○D4.2D14, C4.126(D4×D7), C4.10(C2×D28), Dic14⋊C4⋊8C2, (C4×C28)⋊12C22, C8.D14⋊8C2, C7⋊2(D4.9D4), (C22×D7).3D4, C22.30(D4×D7), C14.28C22≀C2, D4⋊8D14.1C2, D4.9D14⋊1C2, C28.46D4⋊2C2, C4.D28⋊10C2, C4.Dic7⋊5C22, (C2×C28).265C23, C4○D28.14C22, (C2×D28).70C22, C2.31(C22⋊D28), (C2×Dic14)⋊14C22, (C7×M4(2))⋊11C22, (C7×C4≀C2)⋊2C2, (C2×C14).27(C2×D4), (C7×C4○D4).6C22, (C2×C4).110(C22×D7), SmallGroup(448,359)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a5, cac-1=a-1b, dad=ab, cbc-1=a4b, bd=db, dcd=c-1 >
Subgroups: 1020 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≀C2, C4.4D4, C8.C22, 2+ 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, D4.9D4, C56⋊C2, Dic28, C4.Dic7, D14⋊C4, D4.D7, C7⋊Q16, C4×C28, C7×M4(2), C2×Dic14, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, Q8⋊2D7, C7×C4○D4, Dic14⋊C4, C28.46D4, C7×C4≀C2, C4.D28, C8.D14, D4.9D14, D4⋊8D14, M4(2)⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.9D4, C2×D28, D4×D7, C22⋊D28, M4(2)⋊D14
►On 112 points
(1 44 51 30 37 105 112 8)(2 9 52 45 38 31 99 106)(3 46 53 32 39 107 100 10)(4 11 54 47 40 33 101 108)(5 48 55 34 41 109 102 12)(6 13 56 49 42 35 103 110)(7 50 43 36 29 111 104 14)(15 77 61 22 91 68 84 98)(16 85 62 78 92 23 71 69)(17 79 63 24 93 70 72 86)(18 87 64 80 94 25 73 57)(19 81 65 26 95 58 74 88)(20 89 66 82 96 27 75 59)(21 83 67 28 97 60 76 90)
(1 37)(3 39)(5 41)(7 29)(9 31)(11 33)(13 35)(15 91)(17 93)(19 95)(21 97)(23 85)(25 87)(27 89)(43 104)(45 106)(47 108)(49 110)(51 112)(53 100)(55 102)(57 80)(59 82)(61 84)(63 72)(65 74)(67 76)(69 78)
(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 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 98)(15 29)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(43 61)(44 60)(45 59)(46 58)(47 57)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(71 103)(72 102)(73 101)(74 100)(75 99)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)
G:=sub<Sym(112)| (1,44,51,30,37,105,112,8)(2,9,52,45,38,31,99,106)(3,46,53,32,39,107,100,10)(4,11,54,47,40,33,101,108)(5,48,55,34,41,109,102,12)(6,13,56,49,42,35,103,110)(7,50,43,36,29,111,104,14)(15,77,61,22,91,68,84,98)(16,85,62,78,92,23,71,69)(17,79,63,24,93,70,72,86)(18,87,64,80,94,25,73,57)(19,81,65,26,95,58,74,88)(20,89,66,82,96,27,75,59)(21,83,67,28,97,60,76,90), (1,37)(3,39)(5,41)(7,29)(9,31)(11,33)(13,35)(15,91)(17,93)(19,95)(21,97)(23,85)(25,87)(27,89)(43,104)(45,106)(47,108)(49,110)(51,112)(53,100)(55,102)(57,80)(59,82)(61,84)(63,72)(65,74)(67,76)(69,78), (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), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,98)(15,29)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(43,61)(44,60)(45,59)(46,58)(47,57)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(71,103)(72,102)(73,101)(74,100)(75,99)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104)>;
G:=Group( (1,44,51,30,37,105,112,8)(2,9,52,45,38,31,99,106)(3,46,53,32,39,107,100,10)(4,11,54,47,40,33,101,108)(5,48,55,34,41,109,102,12)(6,13,56,49,42,35,103,110)(7,50,43,36,29,111,104,14)(15,77,61,22,91,68,84,98)(16,85,62,78,92,23,71,69)(17,79,63,24,93,70,72,86)(18,87,64,80,94,25,73,57)(19,81,65,26,95,58,74,88)(20,89,66,82,96,27,75,59)(21,83,67,28,97,60,76,90), (1,37)(3,39)(5,41)(7,29)(9,31)(11,33)(13,35)(15,91)(17,93)(19,95)(21,97)(23,85)(25,87)(27,89)(43,104)(45,106)(47,108)(49,110)(51,112)(53,100)(55,102)(57,80)(59,82)(61,84)(63,72)(65,74)(67,76)(69,78), (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), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,98)(15,29)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(43,61)(44,60)(45,59)(46,58)(47,57)(48,70)(49,69)(50,68)(51,67)(52,66)(53,65)(54,64)(55,63)(56,62)(71,103)(72,102)(73,101)(74,100)(75,99)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104) );
G=PermutationGroup([[(1,44,51,30,37,105,112,8),(2,9,52,45,38,31,99,106),(3,46,53,32,39,107,100,10),(4,11,54,47,40,33,101,108),(5,48,55,34,41,109,102,12),(6,13,56,49,42,35,103,110),(7,50,43,36,29,111,104,14),(15,77,61,22,91,68,84,98),(16,85,62,78,92,23,71,69),(17,79,63,24,93,70,72,86),(18,87,64,80,94,25,73,57),(19,81,65,26,95,58,74,88),(20,89,66,82,96,27,75,59),(21,83,67,28,97,60,76,90)], [(1,37),(3,39),(5,41),(7,29),(9,31),(11,33),(13,35),(15,91),(17,93),(19,95),(21,97),(23,85),(25,87),(27,89),(43,104),(45,106),(47,108),(49,110),(51,112),(53,100),(55,102),(57,80),(59,82),(61,84),(63,72),(65,74),(67,76),(69,78)], [(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)], [(1,97),(2,96),(3,95),(4,94),(5,93),(6,92),(7,91),(8,90),(9,89),(10,88),(11,87),(12,86),(13,85),(14,98),(15,29),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(43,61),(44,60),(45,59),(46,58),(47,57),(48,70),(49,69),(50,68),(51,67),(52,66),(53,65),(54,64),(55,63),(56,62),(71,103),(72,102),(73,101),(74,100),(75,99),(76,112),(77,111),(78,110),(79,109),(80,108),(81,107),(82,106),(83,105),(84,104)]])
58 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 | ··· | 28U | 28V | 28W | 28X | 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 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 28 | 56 | 2 | 2 | 2 | 8 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
58 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | D4.9D4 | D4×D7 | D4×D7 | M4(2)⋊D14 |
| kernel | M4(2)⋊D14 | Dic14⋊C4 | C28.46D4 | C7×C4≀C2 | C4.D28 | C8.D14 | D4.9D14 | D4⋊8D14 | Dic14 | D28 | C7×D4 | C7×Q8 | C22×D7 | C4≀C2 | C42 | M4(2) | 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 | 12 |
Matrix representation of M4(2)⋊D14 ►in GL4(𝔽113) generated by
| 0 | 112 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 1 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 85 | 0 | 0 |
| 85 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 0 | 85 | 0 | 0 |
| 85 | 0 | 0 | 0 |
G:=sub<GL(4,GF(113))| [0,98,0,0,112,0,0,0,0,0,0,1,0,0,98,0],[112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,112],[0,85,0,0,85,0,0,0,0,0,0,4,0,0,4,0],[0,0,0,85,0,0,85,0,0,4,0,0,4,0,0,0] >;
M4(2)⋊D14 in GAP, Magma, Sage, TeX
M_4(2)\rtimes D_{14} % in TeX
G:=Group("M4(2):D14"); // GroupNames label
G:=SmallGroup(448,359);
// by ID
G=gap.SmallGroup(448,359);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,58,1123,136,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations