metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).25D14, C8.10(C4×D7), C56.28(C2×C4), (C4×D7).50D4, C8.C4⋊3D7, (C2×C8).68D14, C8⋊D7.1C4, C4.211(D4×D7), C56.C4⋊7C2, D14.7(C4⋊C4), C28.370(C2×D4), C22.4(Q8×D7), (C2×Dic7).4Q8, (C22×D7).3Q8, C28.53D4⋊8C2, Dic7.9(C4⋊C4), (C2×C56).40C22, C28.53(C22×C4), (D7×M4(2)).3C2, C7⋊1(M4(2).C4), (C2×C28).309C23, C4.Dic7.13C22, (C7×M4(2)).19C22, C7⋊C8.3(C2×C4), C4.83(C2×C4×D7), C2.18(D7×C4⋊C4), C14.17(C2×C4⋊C4), (C4×D7).8(C2×C4), (C2×C14).2(C2×Q8), (C7×C8.C4)⋊3C2, (C2×C8⋊D7).1C2, (C2×C7⋊C8).77C22, (C2×C4×D7).43C22, (C2×C4).412(C22×D7), SmallGroup(448,427)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).25D14
G = < a,b,c,d | a8=b2=1, c14=d2=a6b, bab=a5, cac-1=a-1b, dad-1=a3b, bc=cb, bd=db, dcd-1=c13 >
Subgroups: 412 in 102 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, Dic7, C28, D14, D14, C2×C14, C8.C4, C8.C4, C2×M4(2), C7⋊C8, C7⋊C8, C56, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, M4(2).C4, C8×D7, C8⋊D7, C8⋊D7, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C2×C4×D7, C56.C4, C28.53D4, C7×C8.C4, C2×C8⋊D7, D7×M4(2), M4(2).25D14
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C4×D7, C22×D7, M4(2).C4, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4, M4(2).25D14
►On 112 points
Generators in S112
(1 76 43 90 29 104 15 62)(2 91 44 105 30 63 16 77)(3 106 45 64 31 78 17 92)(4 65 46 79 32 93 18 107)(5 80 47 94 33 108 19 66)(6 95 48 109 34 67 20 81)(7 110 49 68 35 82 21 96)(8 69 50 83 36 97 22 111)(9 84 51 98 37 112 23 70)(10 99 52 57 38 71 24 85)(11 58 53 72 39 86 25 100)(12 73 54 87 40 101 26 59)(13 88 55 102 41 60 27 74)(14 103 56 61 42 75 28 89)
(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(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 42 15 56 29 14 43 28)(2 55 16 13 30 27 44 41)(3 12 17 26 31 40 45 54)(4 25 18 39 32 53 46 11)(5 38 19 52 33 10 47 24)(6 51 20 9 34 23 48 37)(7 8 21 22 35 36 49 50)(57 66 71 80 85 94 99 108)(58 79 72 93 86 107 100 65)(59 92 73 106 87 64 101 78)(60 105 74 63 88 77 102 91)(61 62 75 76 89 90 103 104)(67 84 81 98 95 112 109 70)(68 97 82 111 96 69 110 83)
G:=sub<Sym(112)| (1,76,43,90,29,104,15,62)(2,91,44,105,30,63,16,77)(3,106,45,64,31,78,17,92)(4,65,46,79,32,93,18,107)(5,80,47,94,33,108,19,66)(6,95,48,109,34,67,20,81)(7,110,49,68,35,82,21,96)(8,69,50,83,36,97,22,111)(9,84,51,98,37,112,23,70)(10,99,52,57,38,71,24,85)(11,58,53,72,39,86,25,100)(12,73,54,87,40,101,26,59)(13,88,55,102,41,60,27,74)(14,103,56,61,42,75,28,89), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,42,15,56,29,14,43,28)(2,55,16,13,30,27,44,41)(3,12,17,26,31,40,45,54)(4,25,18,39,32,53,46,11)(5,38,19,52,33,10,47,24)(6,51,20,9,34,23,48,37)(7,8,21,22,35,36,49,50)(57,66,71,80,85,94,99,108)(58,79,72,93,86,107,100,65)(59,92,73,106,87,64,101,78)(60,105,74,63,88,77,102,91)(61,62,75,76,89,90,103,104)(67,84,81,98,95,112,109,70)(68,97,82,111,96,69,110,83)>;
G:=Group( (1,76,43,90,29,104,15,62)(2,91,44,105,30,63,16,77)(3,106,45,64,31,78,17,92)(4,65,46,79,32,93,18,107)(5,80,47,94,33,108,19,66)(6,95,48,109,34,67,20,81)(7,110,49,68,35,82,21,96)(8,69,50,83,36,97,22,111)(9,84,51,98,37,112,23,70)(10,99,52,57,38,71,24,85)(11,58,53,72,39,86,25,100)(12,73,54,87,40,101,26,59)(13,88,55,102,41,60,27,74)(14,103,56,61,42,75,28,89), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,42,15,56,29,14,43,28)(2,55,16,13,30,27,44,41)(3,12,17,26,31,40,45,54)(4,25,18,39,32,53,46,11)(5,38,19,52,33,10,47,24)(6,51,20,9,34,23,48,37)(7,8,21,22,35,36,49,50)(57,66,71,80,85,94,99,108)(58,79,72,93,86,107,100,65)(59,92,73,106,87,64,101,78)(60,105,74,63,88,77,102,91)(61,62,75,76,89,90,103,104)(67,84,81,98,95,112,109,70)(68,97,82,111,96,69,110,83) );
G=PermutationGroup([[(1,76,43,90,29,104,15,62),(2,91,44,105,30,63,16,77),(3,106,45,64,31,78,17,92),(4,65,46,79,32,93,18,107),(5,80,47,94,33,108,19,66),(6,95,48,109,34,67,20,81),(7,110,49,68,35,82,21,96),(8,69,50,83,36,97,22,111),(9,84,51,98,37,112,23,70),(10,99,52,57,38,71,24,85),(11,58,53,72,39,86,25,100),(12,73,54,87,40,101,26,59),(13,88,55,102,41,60,27,74),(14,103,56,61,42,75,28,89)], [(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(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)], [(1,42,15,56,29,14,43,28),(2,55,16,13,30,27,44,41),(3,12,17,26,31,40,45,54),(4,25,18,39,32,53,46,11),(5,38,19,52,33,10,47,24),(6,51,20,9,34,23,48,37),(7,8,21,22,35,36,49,50),(57,66,71,80,85,94,99,108),(58,79,72,93,86,107,100,65),(59,92,73,106,87,64,101,78),(60,105,74,63,88,77,102,91),(61,62,75,76,89,90,103,104),(67,84,81,98,95,112,109,70),(68,97,82,111,96,69,110,83)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | ··· | 8F | 8G | ··· | 8L | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 14 | 14 | 1 | 1 | 2 | 14 | 14 | 2 | 2 | 2 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D7 | D14 | D14 | C4×D7 | M4(2).C4 | D4×D7 | Q8×D7 | M4(2).25D14 |
| kernel | M4(2).25D14 | C56.C4 | C28.53D4 | C7×C8.C4 | C2×C8⋊D7 | D7×M4(2) | C8⋊D7 | C4×D7 | C2×Dic7 | C22×D7 | C8.C4 | C2×C8 | M4(2) | C8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 3 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of M4(2).25D14 ►in GL4(𝔽113) generated by
| 0 | 0 | 43 | 13 |
| 0 | 0 | 100 | 70 |
| 70 | 100 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 30 | 75 | 0 | 0 |
| 38 | 38 | 0 | 0 |
| 0 | 0 | 2 | 5 |
| 0 | 0 | 108 | 108 |
| 75 | 30 | 0 | 0 |
| 38 | 38 | 0 | 0 |
| 0 | 0 | 5 | 2 |
| 0 | 0 | 108 | 108 |
G:=sub<GL(4,GF(113))| [0,0,70,13,0,0,100,43,43,100,0,0,13,70,0,0],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[30,38,0,0,75,38,0,0,0,0,2,108,0,0,5,108],[75,38,0,0,30,38,0,0,0,0,5,108,0,0,2,108] >;
M4(2).25D14 in GAP, Magma, Sage, TeX
M_4(2)._{25}D_{14} % in TeX
G:=Group("M4(2).25D14"); // GroupNames label
G:=SmallGroup(448,427);
// by ID
G=gap.SmallGroup(448,427);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,219,58,136,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^14=d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a^3*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^13>;
// generators/relations