metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).19D14, (C4×D7).35D4, C28.90(C2×D4), C4.147(D4×D7), C4.D4⋊6D7, C23.7(C4×D7), C28.D4⋊4C2, (D7×M4(2))⋊6C2, (C2×C28).2C23, (C2×D4).122D14, C4.12D28⋊5C2, D14.2(C22⋊C4), (D4×C14).12C22, (C22×Dic7).4C4, C4.Dic7.1C22, (C7×M4(2)).9C22, Dic7.16(C22⋊C4), (C2×Dic14).44C22, C7⋊1(M4(2).8C22), (C2×C7⋊D4).2C4, (C2×C4×D7).2C22, C22.15(C2×C4×D7), (C7×C4.D4)⋊6C2, C2.14(D7×C22⋊C4), (C2×C4).2(C22×D7), (C2×D4⋊2D7).2C2, C14.13(C2×C22⋊C4), (C22×C14).7(C2×C4), (C2×C14).9(C22×C4), (C2×Dic7).2(C2×C4), (C22×D7).15(C2×C4), SmallGroup(448,279)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).19D14
G = < a,b,c,d | a8=b2=c14=1, d2=a4, bab=a5, cac-1=ab, dad-1=a5b, bc=cb, bd=db, dcd-1=a4c-1 >
Subgroups: 716 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C4.D4, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, M4(2).8C22, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, C4.12D28, C28.D4, C7×C4.D4, D7×M4(2), C2×D4⋊2D7, M4(2).19D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, C22×D7, M4(2).8C22, C2×C4×D7, D4×D7, D7×C22⋊C4, M4(2).19D14
►On 112 points
Generators in S112
(1 98 48 19 70 77 111 37)(2 78 112 20 57 85 49 38)(3 86 50 21 58 79 99 39)(4 80 100 22 59 87 51 40)(5 88 52 23 60 81 101 41)(6 82 102 24 61 89 53 42)(7 90 54 25 62 83 103 29)(8 84 104 26 63 91 55 30)(9 92 56 27 64 71 105 31)(10 72 106 28 65 93 43 32)(11 94 44 15 66 73 107 33)(12 74 108 16 67 95 45 34)(13 96 46 17 68 75 109 35)(14 76 110 18 69 97 47 36)
(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 29)(26 30)(27 31)(28 32)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(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 70 69)(2 68 57 13)(3 12 58 67)(4 66 59 11)(5 10 60 65)(6 64 61 9)(7 8 62 63)(15 40 33 22)(16 21 34 39)(17 38 35 20)(18 19 36 37)(23 32 41 28)(24 27 42 31)(25 30 29 26)(43 101 106 52)(44 51 107 100)(45 99 108 50)(46 49 109 112)(47 111 110 48)(53 105 102 56)(54 55 103 104)(71 89 92 82)(72 81 93 88)(73 87 94 80)(74 79 95 86)(75 85 96 78)(76 77 97 98)(83 91 90 84)
G:=sub<Sym(112)| (1,98,48,19,70,77,111,37)(2,78,112,20,57,85,49,38)(3,86,50,21,58,79,99,39)(4,80,100,22,59,87,51,40)(5,88,52,23,60,81,101,41)(6,82,102,24,61,89,53,42)(7,90,54,25,62,83,103,29)(8,84,104,26,63,91,55,30)(9,92,56,27,64,71,105,31)(10,72,106,28,65,93,43,32)(11,94,44,15,66,73,107,33)(12,74,108,16,67,95,45,34)(13,96,46,17,68,75,109,35)(14,76,110,18,69,97,47,36), (15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,29)(26,30)(27,31)(28,32)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (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), (1,14,70,69)(2,68,57,13)(3,12,58,67)(4,66,59,11)(5,10,60,65)(6,64,61,9)(7,8,62,63)(15,40,33,22)(16,21,34,39)(17,38,35,20)(18,19,36,37)(23,32,41,28)(24,27,42,31)(25,30,29,26)(43,101,106,52)(44,51,107,100)(45,99,108,50)(46,49,109,112)(47,111,110,48)(53,105,102,56)(54,55,103,104)(71,89,92,82)(72,81,93,88)(73,87,94,80)(74,79,95,86)(75,85,96,78)(76,77,97,98)(83,91,90,84)>;
G:=Group( (1,98,48,19,70,77,111,37)(2,78,112,20,57,85,49,38)(3,86,50,21,58,79,99,39)(4,80,100,22,59,87,51,40)(5,88,52,23,60,81,101,41)(6,82,102,24,61,89,53,42)(7,90,54,25,62,83,103,29)(8,84,104,26,63,91,55,30)(9,92,56,27,64,71,105,31)(10,72,106,28,65,93,43,32)(11,94,44,15,66,73,107,33)(12,74,108,16,67,95,45,34)(13,96,46,17,68,75,109,35)(14,76,110,18,69,97,47,36), (15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,29)(26,30)(27,31)(28,32)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (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), (1,14,70,69)(2,68,57,13)(3,12,58,67)(4,66,59,11)(5,10,60,65)(6,64,61,9)(7,8,62,63)(15,40,33,22)(16,21,34,39)(17,38,35,20)(18,19,36,37)(23,32,41,28)(24,27,42,31)(25,30,29,26)(43,101,106,52)(44,51,107,100)(45,99,108,50)(46,49,109,112)(47,111,110,48)(53,105,102,56)(54,55,103,104)(71,89,92,82)(72,81,93,88)(73,87,94,80)(74,79,95,86)(75,85,96,78)(76,77,97,98)(83,91,90,84) );
G=PermutationGroup([[(1,98,48,19,70,77,111,37),(2,78,112,20,57,85,49,38),(3,86,50,21,58,79,99,39),(4,80,100,22,59,87,51,40),(5,88,52,23,60,81,101,41),(6,82,102,24,61,89,53,42),(7,90,54,25,62,83,103,29),(8,84,104,26,63,91,55,30),(9,92,56,27,64,71,105,31),(10,72,106,28,65,93,43,32),(11,94,44,15,66,73,107,33),(12,74,108,16,67,95,45,34),(13,96,46,17,68,75,109,35),(14,76,110,18,69,97,47,36)], [(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,29),(26,30),(27,31),(28,32),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)], [(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)], [(1,14,70,69),(2,68,57,13),(3,12,58,67),(4,66,59,11),(5,10,60,65),(6,64,61,9),(7,8,62,63),(15,40,33,22),(16,21,34,39),(17,38,35,20),(18,19,36,37),(23,32,41,28),(24,27,42,31),(25,30,29,26),(43,101,106,52),(44,51,107,100),(45,99,108,50),(46,49,109,112),(47,111,110,48),(53,105,102,56),(54,55,103,104),(71,89,92,82),(72,81,93,88),(73,87,94,80),(74,79,95,86),(75,85,96,78),(76,77,97,98),(83,91,90,84)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 2 | 2 | 7 | 7 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | D14 | D14 | C4×D7 | M4(2).8C22 | D4×D7 | M4(2).19D14 |
| kernel | M4(2).19D14 | C4.12D28 | C28.D4 | C7×C4.D4 | D7×M4(2) | C2×D4⋊2D7 | C22×Dic7 | C2×C7⋊D4 | C4×D7 | C4.D4 | M4(2) | C2×D4 | C23 | C7 | C4 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 3 | 6 | 3 | 12 | 2 | 6 | 3 |
Matrix representation of M4(2).19D14 ►in GL8(𝔽113)
| 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 112 | 112 | 111 |
| 0 | 0 | 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 8 | 16 | 17 |
| 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 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 96 | 0 | 112 | 112 |
| 0 | 80 | 0 | 80 | 0 | 0 | 0 | 0 |
| 80 | 0 | 80 | 0 | 0 | 0 | 0 | 0 |
| 0 | 33 | 0 | 9 | 0 | 0 | 0 | 0 |
| 33 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 96 | 112 | 112 | 111 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 105 | 1 |
| 0 | 33 | 0 | 33 | 0 | 0 | 0 | 0 |
| 33 | 0 | 33 | 0 | 0 | 0 | 0 | 0 |
| 0 | 104 | 0 | 80 | 0 | 0 | 0 | 0 |
| 104 | 0 | 80 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 17 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 104 | 112 | 8 | 112 |
G:=sub<GL(8,GF(113))| [15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,96,0,0,24,0,0,0,0,112,0,1,8,0,0,0,0,112,98,0,16,0,0,0,0,111,0,0,17],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,96,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,112,0,0,0,0,0,0,0,112],[0,80,0,33,0,0,0,0,80,0,33,0,0,0,0,0,0,80,0,9,0,0,0,0,80,0,9,0,0,0,0,0,0,0,0,0,0,96,1,8,0,0,0,0,0,112,0,0,0,0,0,0,1,112,0,105,0,0,0,0,0,111,0,1],[0,33,0,104,0,0,0,0,33,0,104,0,0,0,0,0,0,33,0,80,0,0,0,0,33,0,80,0,0,0,0,0,0,0,0,0,0,17,1,104,0,0,0,0,0,1,0,112,0,0,0,0,112,1,0,8,0,0,0,0,0,2,0,112] >;
M4(2).19D14 in GAP, Magma, Sage, TeX
M_4(2)._{19}D_{14} % in TeX
G:=Group("M4(2).19D14"); // GroupNames label
G:=SmallGroup(448,279);
// by ID
G=gap.SmallGroup(448,279);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,219,58,570,136,438,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d^-1=a^5*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations