metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).12D14, C7⋊C8.30D4, C8⋊C22⋊2D7, (C7×D4).11D4, C4.178(D4×D7), (C7×Q8).11D4, D4⋊D14⋊4C2, C4○D4.24D14, (C2×D4).79D14, C28.195(C2×D4), C28.D4⋊9C2, C7⋊5(D4.4D4), D4.4(C7⋊D4), Q8.Dic7⋊5C2, Q8.4(C7⋊D4), C28.53D4⋊9C2, (C2×C28).14C23, C28.46D4⋊10C2, C14.124(C4⋊D4), (D4×C14).104C22, (C2×D28).129C22, C4.Dic7.24C22, C2.30(Dic7⋊D4), C22.13(D4⋊2D7), (C7×M4(2)).22C22, (C2×D4⋊D7)⋊22C2, (C7×C8⋊C22)⋊6C2, C4.51(C2×C7⋊D4), (C2×C7⋊C8).170C22, (C2×C4).14(C22×D7), (C2×C14).36(C4○D4), (C7×C4○D4).12C22, SmallGroup(448,733)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).D14
G = < a,b,c,d | a8=b2=c14=1, d2=a6, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >
Subgroups: 556 in 108 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C28, C28, D14, C2×C14, C2×C14, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C7⋊C8, C7⋊C8, C56, D28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×C14, D4.4D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4⋊D7, Q8⋊D7, C7×M4(2), C7×D8, C7×SD16, C2×D28, D4×C14, C7×C4○D4, C28.53D4, C28.46D4, C28.D4, C2×D4⋊D7, Q8.Dic7, D4⋊D14, C7×C8⋊C22, M4(2).D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.4D4, D4×D7, D4⋊2D7, C2×C7⋊D4, Dic7⋊D4, M4(2).D14
►On 112 points
Generators in S112
(1 111 23 54 90 63 33 82)(2 83 34 64 91 55 24 112)(3 99 25 56 92 65 35 84)(4 71 36 66 93 43 26 100)(5 101 27 44 94 67 37 72)(6 73 38 68 95 45 28 102)(7 103 15 46 96 69 39 74)(8 75 40 70 97 47 16 104)(9 105 17 48 98 57 41 76)(10 77 42 58 85 49 18 106)(11 107 19 50 86 59 29 78)(12 79 30 60 87 51 20 108)(13 109 21 52 88 61 31 80)(14 81 32 62 89 53 22 110)
(1 90)(3 92)(5 94)(7 96)(9 98)(11 86)(13 88)(15 39)(17 41)(19 29)(21 31)(23 33)(25 35)(27 37)(43 71)(45 73)(47 75)(49 77)(51 79)(53 81)(55 83)(58 106)(60 108)(62 110)(64 112)(66 100)(68 102)(70 104)
(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 14 33 22 90 89 23 32)(2 31 24 88 91 21 34 13)(3 12 35 20 92 87 25 30)(4 29 26 86 93 19 36 11)(5 10 37 18 94 85 27 42)(6 41 28 98 95 17 38 9)(7 8 39 16 96 97 15 40)(43 59 66 50 71 107 100 78)(44 77 101 106 72 49 67 58)(45 57 68 48 73 105 102 76)(46 75 103 104 74 47 69 70)(51 65 60 56 79 99 108 84)(52 83 109 112 80 55 61 64)(53 63 62 54 81 111 110 82)
G:=sub<Sym(112)| (1,111,23,54,90,63,33,82)(2,83,34,64,91,55,24,112)(3,99,25,56,92,65,35,84)(4,71,36,66,93,43,26,100)(5,101,27,44,94,67,37,72)(6,73,38,68,95,45,28,102)(7,103,15,46,96,69,39,74)(8,75,40,70,97,47,16,104)(9,105,17,48,98,57,41,76)(10,77,42,58,85,49,18,106)(11,107,19,50,86,59,29,78)(12,79,30,60,87,51,20,108)(13,109,21,52,88,61,31,80)(14,81,32,62,89,53,22,110), (1,90)(3,92)(5,94)(7,96)(9,98)(11,86)(13,88)(15,39)(17,41)(19,29)(21,31)(23,33)(25,35)(27,37)(43,71)(45,73)(47,75)(49,77)(51,79)(53,81)(55,83)(58,106)(60,108)(62,110)(64,112)(66,100)(68,102)(70,104), (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), (1,14,33,22,90,89,23,32)(2,31,24,88,91,21,34,13)(3,12,35,20,92,87,25,30)(4,29,26,86,93,19,36,11)(5,10,37,18,94,85,27,42)(6,41,28,98,95,17,38,9)(7,8,39,16,96,97,15,40)(43,59,66,50,71,107,100,78)(44,77,101,106,72,49,67,58)(45,57,68,48,73,105,102,76)(46,75,103,104,74,47,69,70)(51,65,60,56,79,99,108,84)(52,83,109,112,80,55,61,64)(53,63,62,54,81,111,110,82)>;
G:=Group( (1,111,23,54,90,63,33,82)(2,83,34,64,91,55,24,112)(3,99,25,56,92,65,35,84)(4,71,36,66,93,43,26,100)(5,101,27,44,94,67,37,72)(6,73,38,68,95,45,28,102)(7,103,15,46,96,69,39,74)(8,75,40,70,97,47,16,104)(9,105,17,48,98,57,41,76)(10,77,42,58,85,49,18,106)(11,107,19,50,86,59,29,78)(12,79,30,60,87,51,20,108)(13,109,21,52,88,61,31,80)(14,81,32,62,89,53,22,110), (1,90)(3,92)(5,94)(7,96)(9,98)(11,86)(13,88)(15,39)(17,41)(19,29)(21,31)(23,33)(25,35)(27,37)(43,71)(45,73)(47,75)(49,77)(51,79)(53,81)(55,83)(58,106)(60,108)(62,110)(64,112)(66,100)(68,102)(70,104), (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), (1,14,33,22,90,89,23,32)(2,31,24,88,91,21,34,13)(3,12,35,20,92,87,25,30)(4,29,26,86,93,19,36,11)(5,10,37,18,94,85,27,42)(6,41,28,98,95,17,38,9)(7,8,39,16,96,97,15,40)(43,59,66,50,71,107,100,78)(44,77,101,106,72,49,67,58)(45,57,68,48,73,105,102,76)(46,75,103,104,74,47,69,70)(51,65,60,56,79,99,108,84)(52,83,109,112,80,55,61,64)(53,63,62,54,81,111,110,82) );
G=PermutationGroup([[(1,111,23,54,90,63,33,82),(2,83,34,64,91,55,24,112),(3,99,25,56,92,65,35,84),(4,71,36,66,93,43,26,100),(5,101,27,44,94,67,37,72),(6,73,38,68,95,45,28,102),(7,103,15,46,96,69,39,74),(8,75,40,70,97,47,16,104),(9,105,17,48,98,57,41,76),(10,77,42,58,85,49,18,106),(11,107,19,50,86,59,29,78),(12,79,30,60,87,51,20,108),(13,109,21,52,88,61,31,80),(14,81,32,62,89,53,22,110)], [(1,90),(3,92),(5,94),(7,96),(9,98),(11,86),(13,88),(15,39),(17,41),(19,29),(21,31),(23,33),(25,35),(27,37),(43,71),(45,73),(47,75),(49,77),(51,79),(53,81),(55,83),(58,106),(60,108),(62,110),(64,112),(66,100),(68,102),(70,104)], [(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)], [(1,14,33,22,90,89,23,32),(2,31,24,88,91,21,34,13),(3,12,35,20,92,87,25,30),(4,29,26,86,93,19,36,11),(5,10,37,18,94,85,27,42),(6,41,28,98,95,17,38,9),(7,8,39,16,96,97,15,40),(43,59,66,50,71,107,100,78),(44,77,101,106,72,49,67,58),(45,57,68,48,73,105,102,76),(46,75,103,104,74,47,69,70),(51,65,60,56,79,99,108,84),(52,83,109,112,80,55,61,64),(53,63,62,54,81,111,110,82)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 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 | 2 | 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 | 56 | 2 | 2 | 4 | 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 | D4.4D4 | D4×D7 | D4⋊2D7 | M4(2).D14 |
| kernel | M4(2).D14 | C28.53D4 | C28.46D4 | C28.D4 | C2×D4⋊D7 | Q8.Dic7 | D4⋊D14 | 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).D14 ►in GL8(𝔽113)
| 23 | 66 | 44 | 96 | 0 | 0 | 0 | 0 |
| 39 | 11 | 100 | 84 | 0 | 0 | 0 | 0 |
| 52 | 23 | 57 | 84 | 0 | 0 | 0 | 0 |
| 38 | 32 | 17 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 83 | 83 | 51 | 62 |
| 0 | 0 | 0 | 0 | 15 | 15 | 0 | 51 |
| 0 | 0 | 0 | 0 | 82 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 98 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 26 | 1 | 0 |
| 0 | 0 | 0 | 0 | 87 | 87 | 0 | 1 |
| 28 | 5 | 112 | 99 | 0 | 0 | 0 | 0 |
| 61 | 88 | 102 | 12 | 0 | 0 | 0 | 0 |
| 9 | 38 | 42 | 72 | 0 | 0 | 0 | 0 |
| 87 | 79 | 96 | 68 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 87 | 87 | 111 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 55 | 112 | 13 | 100 |
| 0 | 0 | 0 | 0 | 58 | 2 | 100 | 13 |
| 20 | 9 | 14 | 1 | 0 | 0 | 0 | 0 |
| 54 | 35 | 101 | 11 | 0 | 0 | 0 | 0 |
| 2 | 98 | 41 | 71 | 0 | 0 | 0 | 0 |
| 102 | 54 | 45 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 26 | 0 | 111 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 57 | 1 | 100 | 13 |
| 0 | 0 | 0 | 0 | 55 | 111 | 13 | 100 |
G:=sub<GL(8,GF(113))| [23,39,52,38,0,0,0,0,66,11,23,32,0,0,0,0,44,100,57,17,0,0,0,0,96,84,84,22,0,0,0,0,0,0,0,0,83,15,82,0,0,0,0,0,83,15,0,0,0,0,0,0,51,0,15,98,0,0,0,0,62,51,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,26,87,0,0,0,0,0,112,26,87,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,61,9,87,0,0,0,0,5,88,38,79,0,0,0,0,112,102,42,96,0,0,0,0,99,12,72,68,0,0,0,0,0,0,0,0,87,0,55,58,0,0,0,0,87,0,112,2,0,0,0,0,111,1,13,100,0,0,0,0,0,1,100,13],[20,54,2,102,0,0,0,0,9,35,98,54,0,0,0,0,14,101,41,45,0,0,0,0,1,11,71,17,0,0,0,0,0,0,0,0,26,0,57,55,0,0,0,0,26,0,1,111,0,0,0,0,0,1,100,13,0,0,0,0,111,1,13,100] >;
M4(2).D14 in GAP, Magma, Sage, TeX
M_4(2).D_{14} % in TeX
G:=Group("M4(2).D14"); // GroupNames label
G:=SmallGroup(448,733);
// by ID
G=gap.SmallGroup(448,733);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,219,1123,297,136,1684,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^6,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