metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).13D14, C7⋊C8.31D4, C8⋊C22.D7, (C7×D4).12D4, C4.179(D4×D7), (C7×Q8).12D4, C4○D4.25D14, (C2×D4).80D14, C28.196(C2×D4), C7⋊6(D4.3D4), D4.5(C7⋊D4), Q8.Dic7⋊6C2, Q8.5(C7⋊D4), D4.9D14⋊4C2, C4.12D28⋊9C2, C28.D4⋊10C2, (C2×C28).15C23, C28.53D4⋊10C2, C14.125(C4⋊D4), (D4×C14).105C22, C4.Dic7.25C22, C2.31(Dic7⋊D4), C22.14(D4⋊2D7), (C7×M4(2)).23C22, (C2×Dic14).134C22, C4.52(C2×C7⋊D4), (C2×D4.D7)⋊22C2, (C7×C8⋊C22).1C2, (C2×C7⋊C8).171C22, (C2×C4).15(C22×D7), (C2×C14).37(C4○D4), (C7×C4○D4).13C22, SmallGroup(448,734)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).13D14
G = < a,b,c,d | a8=b2=c14=1, d2=a2, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >
Subgroups: 428 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C7⋊C8, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4.3D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4.D7, C7⋊Q16, C7×M4(2), C7×D8, C7×SD16, C2×Dic14, D4×C14, C7×C4○D4, C28.53D4, C4.12D28, C28.D4, C2×D4.D7, Q8.Dic7, D4.9D14, C7×C8⋊C22, M4(2).13D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.3D4, D4×D7, D4⋊2D7, C2×C7⋊D4, Dic7⋊D4, M4(2).13D14
►On 112 points
Generators in S112
(1 8 66 55 106 99 48 59)(2 60 49 100 107 56 67 9)(3 10 68 43 108 101 50 61)(4 62 51 102 109 44 69 11)(5 12 70 45 110 103 52 63)(6 64 53 104 111 46 57 13)(7 14 58 47 112 105 54 65)(15 73 80 22 88 31 38 95)(16 96 39 32 89 23 81 74)(17 75 82 24 90 33 40 97)(18 98 41 34 91 25 83 76)(19 77 84 26 92 35 42 85)(20 86 29 36 93 27 71 78)(21 79 72 28 94 37 30 87)
(1 106)(3 108)(5 110)(7 112)(9 100)(11 102)(13 104)(15 88)(17 90)(19 92)(21 94)(23 96)(25 98)(27 86)(30 72)(32 74)(34 76)(36 78)(38 80)(40 82)(42 84)(44 62)(46 64)(48 66)(50 68)(52 70)(54 58)(56 60)
(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 39 66 89 106 81 48 16)(2 88 49 38 107 15 67 80)(3 37 68 87 108 79 50 28)(4 86 51 36 109 27 69 78)(5 35 70 85 110 77 52 26)(6 98 53 34 111 25 57 76)(7 33 58 97 112 75 54 24)(8 96 55 32 99 23 59 74)(9 31 60 95 100 73 56 22)(10 94 43 30 101 21 61 72)(11 29 62 93 102 71 44 20)(12 92 45 42 103 19 63 84)(13 41 64 91 104 83 46 18)(14 90 47 40 105 17 65 82)
G:=sub<Sym(112)| (1,8,66,55,106,99,48,59)(2,60,49,100,107,56,67,9)(3,10,68,43,108,101,50,61)(4,62,51,102,109,44,69,11)(5,12,70,45,110,103,52,63)(6,64,53,104,111,46,57,13)(7,14,58,47,112,105,54,65)(15,73,80,22,88,31,38,95)(16,96,39,32,89,23,81,74)(17,75,82,24,90,33,40,97)(18,98,41,34,91,25,83,76)(19,77,84,26,92,35,42,85)(20,86,29,36,93,27,71,78)(21,79,72,28,94,37,30,87), (1,106)(3,108)(5,110)(7,112)(9,100)(11,102)(13,104)(15,88)(17,90)(19,92)(21,94)(23,96)(25,98)(27,86)(30,72)(32,74)(34,76)(36,78)(38,80)(40,82)(42,84)(44,62)(46,64)(48,66)(50,68)(52,70)(54,58)(56,60), (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), (1,39,66,89,106,81,48,16)(2,88,49,38,107,15,67,80)(3,37,68,87,108,79,50,28)(4,86,51,36,109,27,69,78)(5,35,70,85,110,77,52,26)(6,98,53,34,111,25,57,76)(7,33,58,97,112,75,54,24)(8,96,55,32,99,23,59,74)(9,31,60,95,100,73,56,22)(10,94,43,30,101,21,61,72)(11,29,62,93,102,71,44,20)(12,92,45,42,103,19,63,84)(13,41,64,91,104,83,46,18)(14,90,47,40,105,17,65,82)>;
G:=Group( (1,8,66,55,106,99,48,59)(2,60,49,100,107,56,67,9)(3,10,68,43,108,101,50,61)(4,62,51,102,109,44,69,11)(5,12,70,45,110,103,52,63)(6,64,53,104,111,46,57,13)(7,14,58,47,112,105,54,65)(15,73,80,22,88,31,38,95)(16,96,39,32,89,23,81,74)(17,75,82,24,90,33,40,97)(18,98,41,34,91,25,83,76)(19,77,84,26,92,35,42,85)(20,86,29,36,93,27,71,78)(21,79,72,28,94,37,30,87), (1,106)(3,108)(5,110)(7,112)(9,100)(11,102)(13,104)(15,88)(17,90)(19,92)(21,94)(23,96)(25,98)(27,86)(30,72)(32,74)(34,76)(36,78)(38,80)(40,82)(42,84)(44,62)(46,64)(48,66)(50,68)(52,70)(54,58)(56,60), (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), (1,39,66,89,106,81,48,16)(2,88,49,38,107,15,67,80)(3,37,68,87,108,79,50,28)(4,86,51,36,109,27,69,78)(5,35,70,85,110,77,52,26)(6,98,53,34,111,25,57,76)(7,33,58,97,112,75,54,24)(8,96,55,32,99,23,59,74)(9,31,60,95,100,73,56,22)(10,94,43,30,101,21,61,72)(11,29,62,93,102,71,44,20)(12,92,45,42,103,19,63,84)(13,41,64,91,104,83,46,18)(14,90,47,40,105,17,65,82) );
G=PermutationGroup([[(1,8,66,55,106,99,48,59),(2,60,49,100,107,56,67,9),(3,10,68,43,108,101,50,61),(4,62,51,102,109,44,69,11),(5,12,70,45,110,103,52,63),(6,64,53,104,111,46,57,13),(7,14,58,47,112,105,54,65),(15,73,80,22,88,31,38,95),(16,96,39,32,89,23,81,74),(17,75,82,24,90,33,40,97),(18,98,41,34,91,25,83,76),(19,77,84,26,92,35,42,85),(20,86,29,36,93,27,71,78),(21,79,72,28,94,37,30,87)], [(1,106),(3,108),(5,110),(7,112),(9,100),(11,102),(13,104),(15,88),(17,90),(19,92),(21,94),(23,96),(25,98),(27,86),(30,72),(32,74),(34,76),(36,78),(38,80),(40,82),(42,84),(44,62),(46,64),(48,66),(50,68),(52,70),(54,58),(56,60)], [(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)], [(1,39,66,89,106,81,48,16),(2,88,49,38,107,15,67,80),(3,37,68,87,108,79,50,28),(4,86,51,36,109,27,69,78),(5,35,70,85,110,77,52,26),(6,98,53,34,111,25,57,76),(7,33,58,97,112,75,54,24),(8,96,55,32,99,23,59,74),(9,31,60,95,100,73,56,22),(10,94,43,30,101,21,61,72),(11,29,62,93,102,71,44,20),(12,92,45,42,103,19,63,84),(13,41,64,91,104,83,46,18),(14,90,47,40,105,17,65,82)]])
49 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 | 14E | 14F | 14G | ··· | 14O | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56F |
| 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 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 8 | 2 | 2 | 4 | 56 | 2 | 2 | 2 | 8 | 14 | 14 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | 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 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | D4.3D4 | D4×D7 | D4⋊2D7 | M4(2).13D14 |
| kernel | M4(2).13D14 | C28.53D4 | C4.12D28 | C28.D4 | C2×D4.D7 | Q8.Dic7 | D4.9D14 | C7×C8⋊C22 | C7⋊C8 | C7×D4 | C7×Q8 | C8⋊C22 | C2×C14 | M4(2) | C2×D4 | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 3 |
Matrix representation of M4(2).13D14 ►in GL8(𝔽113)
| 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 | 0 | 1 | 77 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 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 | 1 |
| 78 | 0 | 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 35 | 0 | 88 | 0 | 0 | 0 | 0 |
| 63 | 0 | 26 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 87 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 48 | 0 | 54 | 0 | 0 | 0 | 0 |
| 65 | 0 | 59 | 0 | 0 | 0 | 0 | 0 |
| 0 | 85 | 0 | 65 | 0 | 0 | 0 | 0 |
| 28 | 0 | 48 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 26 | 97 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 87 | 0 | 0 |
G:=sub<GL(8,GF(113))| [0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,1,44,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,77,112,0,0],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[78,0,63,0,0,0,0,0,0,35,0,50,0,0,0,0,25,0,26,0,0,0,0,0,0,88,0,87,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,65,0,28,0,0,0,0,48,0,85,0,0,0,0,0,0,59,0,48,0,0,0,0,54,0,65,0,0,0,0,0,0,0,0,0,0,0,26,7,0,0,0,0,0,0,97,87,0,0,0,0,0,7,0,0,0,0,0,0,16,0,0,0] >;
M4(2).13D14 in GAP, Magma, Sage, TeX
M_4(2)._{13}D_{14} % in TeX
G:=Group("M4(2).13D14"); // GroupNames label
G:=SmallGroup(448,734);
// by ID
G=gap.SmallGroup(448,734);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,219,1123,297,136,1684,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^2,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations