metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).15D14, C7⋊C8.32D4, (C7×D4).15D4, C4.180(D4×D7), (C7×Q8).15D4, C8.C22⋊2D7, C4○D4.26D14, C28.199(C2×D4), C7⋊7(D4.3D4), D4.6(C7⋊D4), Q8.Dic7⋊7C2, (C2×Q8).69D14, Q8.6(C7⋊D4), D4⋊D14.2C2, C28.10D4⋊9C2, (C2×C28).18C23, C28.46D4⋊12C2, C28.53D4⋊11C2, (Q8×C14).96C22, C14.126(C4⋊D4), (C2×D28).131C22, C4.Dic7.27C22, C2.32(Dic7⋊D4), C22.15(D4⋊2D7), (C7×M4(2)).25C22, (C2×Q8⋊D7)⋊22C2, C4.55(C2×C7⋊D4), (C7×C8.C22)⋊6C2, (C2×C7⋊C8).172C22, (C2×C4).18(C22×D7), (C2×C14).38(C4○D4), (C7×C4○D4).16C22, SmallGroup(448,737)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).15D14
G = < a,b,c,d | a8=b2=c14=1, d2=a2, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >
Subgroups: 524 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, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C7⋊C8, C56, D28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, D4.3D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4⋊D7, Q8⋊D7, C7×M4(2), C7×SD16, C7×Q16, C2×D28, Q8×C14, C7×C4○D4, C28.53D4, C28.46D4, C28.10D4, C2×Q8⋊D7, Q8.Dic7, D4⋊D14, C7×C8.C22, M4(2).15D14
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).15D14
►On 112 points
Generators in S112
(1 63 41 77 53 95 108 22)(2 78 109 64 54 23 42 96)(3 65 29 79 55 97 110 24)(4 80 111 66 56 25 30 98)(5 67 31 81 43 85 112 26)(6 82 99 68 44 27 32 86)(7 69 33 83 45 87 100 28)(8 84 101 70 46 15 34 88)(9 57 35 71 47 89 102 16)(10 72 103 58 48 17 36 90)(11 59 37 73 49 91 104 18)(12 74 105 60 50 19 38 92)(13 61 39 75 51 93 106 20)(14 76 107 62 52 21 40 94)
(2 54)(4 56)(6 44)(8 46)(10 48)(12 50)(14 52)(16 71)(18 73)(20 75)(22 77)(24 79)(26 81)(28 83)(30 111)(32 99)(34 101)(36 103)(38 105)(40 107)(42 109)(57 89)(59 91)(61 93)(63 95)(65 97)(67 85)(69 87)
(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 41 107 53 52 108 40)(2 39 109 51 54 106 42 13)(3 12 29 105 55 50 110 38)(4 37 111 49 56 104 30 11)(5 10 31 103 43 48 112 36)(6 35 99 47 44 102 32 9)(7 8 33 101 45 46 100 34)(15 87 88 28 84 69 70 83)(16 82 57 68 71 27 89 86)(17 85 90 26 72 67 58 81)(18 80 59 66 73 25 91 98)(19 97 92 24 74 65 60 79)(20 78 61 64 75 23 93 96)(21 95 94 22 76 63 62 77)
G:=sub<Sym(112)| (1,63,41,77,53,95,108,22)(2,78,109,64,54,23,42,96)(3,65,29,79,55,97,110,24)(4,80,111,66,56,25,30,98)(5,67,31,81,43,85,112,26)(6,82,99,68,44,27,32,86)(7,69,33,83,45,87,100,28)(8,84,101,70,46,15,34,88)(9,57,35,71,47,89,102,16)(10,72,103,58,48,17,36,90)(11,59,37,73,49,91,104,18)(12,74,105,60,50,19,38,92)(13,61,39,75,51,93,106,20)(14,76,107,62,52,21,40,94), (2,54)(4,56)(6,44)(8,46)(10,48)(12,50)(14,52)(16,71)(18,73)(20,75)(22,77)(24,79)(26,81)(28,83)(30,111)(32,99)(34,101)(36,103)(38,105)(40,107)(42,109)(57,89)(59,91)(61,93)(63,95)(65,97)(67,85)(69,87), (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), (1,14,41,107,53,52,108,40)(2,39,109,51,54,106,42,13)(3,12,29,105,55,50,110,38)(4,37,111,49,56,104,30,11)(5,10,31,103,43,48,112,36)(6,35,99,47,44,102,32,9)(7,8,33,101,45,46,100,34)(15,87,88,28,84,69,70,83)(16,82,57,68,71,27,89,86)(17,85,90,26,72,67,58,81)(18,80,59,66,73,25,91,98)(19,97,92,24,74,65,60,79)(20,78,61,64,75,23,93,96)(21,95,94,22,76,63,62,77)>;
G:=Group( (1,63,41,77,53,95,108,22)(2,78,109,64,54,23,42,96)(3,65,29,79,55,97,110,24)(4,80,111,66,56,25,30,98)(5,67,31,81,43,85,112,26)(6,82,99,68,44,27,32,86)(7,69,33,83,45,87,100,28)(8,84,101,70,46,15,34,88)(9,57,35,71,47,89,102,16)(10,72,103,58,48,17,36,90)(11,59,37,73,49,91,104,18)(12,74,105,60,50,19,38,92)(13,61,39,75,51,93,106,20)(14,76,107,62,52,21,40,94), (2,54)(4,56)(6,44)(8,46)(10,48)(12,50)(14,52)(16,71)(18,73)(20,75)(22,77)(24,79)(26,81)(28,83)(30,111)(32,99)(34,101)(36,103)(38,105)(40,107)(42,109)(57,89)(59,91)(61,93)(63,95)(65,97)(67,85)(69,87), (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), (1,14,41,107,53,52,108,40)(2,39,109,51,54,106,42,13)(3,12,29,105,55,50,110,38)(4,37,111,49,56,104,30,11)(5,10,31,103,43,48,112,36)(6,35,99,47,44,102,32,9)(7,8,33,101,45,46,100,34)(15,87,88,28,84,69,70,83)(16,82,57,68,71,27,89,86)(17,85,90,26,72,67,58,81)(18,80,59,66,73,25,91,98)(19,97,92,24,74,65,60,79)(20,78,61,64,75,23,93,96)(21,95,94,22,76,63,62,77) );
G=PermutationGroup([[(1,63,41,77,53,95,108,22),(2,78,109,64,54,23,42,96),(3,65,29,79,55,97,110,24),(4,80,111,66,56,25,30,98),(5,67,31,81,43,85,112,26),(6,82,99,68,44,27,32,86),(7,69,33,83,45,87,100,28),(8,84,101,70,46,15,34,88),(9,57,35,71,47,89,102,16),(10,72,103,58,48,17,36,90),(11,59,37,73,49,91,104,18),(12,74,105,60,50,19,38,92),(13,61,39,75,51,93,106,20),(14,76,107,62,52,21,40,94)], [(2,54),(4,56),(6,44),(8,46),(10,48),(12,50),(14,52),(16,71),(18,73),(20,75),(22,77),(24,79),(26,81),(28,83),(30,111),(32,99),(34,101),(36,103),(38,105),(40,107),(42,109),(57,89),(59,91),(61,93),(63,95),(65,97),(67,85),(69,87)], [(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)], [(1,14,41,107,53,52,108,40),(2,39,109,51,54,106,42,13),(3,12,29,105,55,50,110,38),(4,37,111,49,56,104,30,11),(5,10,31,103,43,48,112,36),(6,35,99,47,44,102,32,9),(7,8,33,101,45,46,100,34),(15,87,88,28,84,69,70,83),(16,82,57,68,71,27,89,86),(17,85,90,26,72,67,58,81),(18,80,59,66,73,25,91,98),(19,97,92,24,74,65,60,79),(20,78,61,64,75,23,93,96),(21,95,94,22,76,63,62,77)]])
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 | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 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 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 56 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 | 14 | 14 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 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).15D14 |
| kernel | M4(2).15D14 | C28.53D4 | C28.46D4 | C28.10D4 | C2×Q8⋊D7 | Q8.Dic7 | D4⋊D14 | C7×C8.C22 | C7⋊C8 | C7×D4 | C7×Q8 | C8.C22 | C2×C14 | M4(2) | C2×Q8 | 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).15D14 ►in GL6(𝔽113)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 87 |
| 0 | 0 | 2 | 1 | 7 | 0 |
| 0 | 0 | 0 | 0 | 112 | 72 |
| 0 | 0 | 0 | 91 | 0 | 98 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 32 | 112 | 0 |
| 0 | 0 | 62 | 0 | 0 | 112 |
| 101 | 16 | 0 | 0 | 0 | 0 |
| 16 | 101 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 91 | 72 | 0 |
| 0 | 0 | 88 | 0 | 0 | 81 |
| 0 | 0 | 11 | 87 | 44 | 53 |
| 0 | 0 | 105 | 56 | 85 | 0 |
| 101 | 16 | 0 | 0 | 0 | 0 |
| 97 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 91 | 72 | 111 |
| 0 | 0 | 25 | 0 | 0 | 32 |
| 0 | 0 | 83 | 87 | 44 | 109 |
| 0 | 0 | 107 | 49 | 85 | 51 |
G:=sub<GL(6,GF(113))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,15,2,0,0,0,0,0,1,0,91,0,0,0,7,112,0,0,0,87,0,72,98],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,64,62,0,0,0,1,32,0,0,0,0,0,112,0,0,0,0,0,0,112],[101,16,0,0,0,0,16,101,0,0,0,0,0,0,69,88,11,105,0,0,91,0,87,56,0,0,72,0,44,85,0,0,0,81,53,0],[101,97,0,0,0,0,16,12,0,0,0,0,0,0,18,25,83,107,0,0,91,0,87,49,0,0,72,0,44,85,0,0,111,32,109,51] >;
M4(2).15D14 in GAP, Magma, Sage, TeX
M_4(2)._{15}D_{14} % in TeX
G:=Group("M4(2).15D14"); // GroupNames label
G:=SmallGroup(448,737);
// by ID
G=gap.SmallGroup(448,737);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,219,184,1123,297,136,1684,851,438,102,18822]);
// 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^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;
// generators/relations