metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).31D14, (C2×C4).50D28, C4.66(C2×D28), (C2×C28).174D4, C28.418(C2×D4), C23.21(C4×D7), C4.56(D14⋊C4), (C2×M4(2))⋊12D7, C28.46D4⋊15C2, C4.12D28⋊15C2, C28.67(C22⋊C4), (C14×M4(2))⋊20C2, (C2×C28).417C23, C22.7(D14⋊C4), (C22×C4).140D14, (C2×D28).251C22, C4.Dic7.42C22, (C22×C28).189C22, (C7×M4(2)).34C22, (C2×Dic14).278C22, C7⋊3(M4(2).8C22), (C2×C4×D7).4C4, C22.21(C2×C4×D7), C2.31(C2×D14⋊C4), (C2×C4).160(C4×D7), (C2×C7⋊D4).13C4, C4.111(C2×C7⋊D4), (C2×C28).109(C2×C4), (C2×C4○D28).12C2, (C2×C4).77(C7⋊D4), C14.59(C2×C22⋊C4), (C2×C4.Dic7)⋊16C2, (C2×Dic7).5(C2×C4), (C22×D7).6(C2×C4), (C22×C14).71(C2×C4), (C2×C14).15(C22×C4), (C2×C4).121(C22×D7), (C2×C14).66(C22⋊C4), SmallGroup(448,666)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).31D14
G = < a,b,c,d | a8=b2=1, c14=d2=a4, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c13 >
Subgroups: 740 in 150 conjugacy classes, 59 normal (41 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, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4.10D4, C2×M4(2), C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, M4(2).8C22, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C7×M4(2), C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, C28.46D4, C4.12D28, C2×C4.Dic7, C14×M4(2), C2×C4○D28, M4(2).31D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, M4(2).8C22, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C2×D14⋊C4, M4(2).31D14
►On 112 points
(1 48 82 98 15 34 68 112)(2 49 83 99 16 35 69 85)(3 50 84 100 17 36 70 86)(4 51 57 101 18 37 71 87)(5 52 58 102 19 38 72 88)(6 53 59 103 20 39 73 89)(7 54 60 104 21 40 74 90)(8 55 61 105 22 41 75 91)(9 56 62 106 23 42 76 92)(10 29 63 107 24 43 77 93)(11 30 64 108 25 44 78 94)(12 31 65 109 26 45 79 95)(13 32 66 110 27 46 80 96)(14 33 67 111 28 47 81 97)
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 81 15 67)(2 66 16 80)(3 79 17 65)(4 64 18 78)(5 77 19 63)(6 62 20 76)(7 75 21 61)(8 60 22 74)(9 73 23 59)(10 58 24 72)(11 71 25 57)(12 84 26 70)(13 69 27 83)(14 82 28 68)(29 88 43 102)(30 101 44 87)(31 86 45 100)(32 99 46 85)(33 112 47 98)(34 97 48 111)(35 110 49 96)(36 95 50 109)(37 108 51 94)(38 93 52 107)(39 106 53 92)(40 91 54 105)(41 104 55 90)(42 89 56 103)
G:=sub<Sym(112)| (1,48,82,98,15,34,68,112)(2,49,83,99,16,35,69,85)(3,50,84,100,17,36,70,86)(4,51,57,101,18,37,71,87)(5,52,58,102,19,38,72,88)(6,53,59,103,20,39,73,89)(7,54,60,104,21,40,74,90)(8,55,61,105,22,41,75,91)(9,56,62,106,23,42,76,92)(10,29,63,107,24,43,77,93)(11,30,64,108,25,44,78,94)(12,31,65,109,26,45,79,95)(13,32,66,110,27,46,80,96)(14,33,67,111,28,47,81,97), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,81,15,67)(2,66,16,80)(3,79,17,65)(4,64,18,78)(5,77,19,63)(6,62,20,76)(7,75,21,61)(8,60,22,74)(9,73,23,59)(10,58,24,72)(11,71,25,57)(12,84,26,70)(13,69,27,83)(14,82,28,68)(29,88,43,102)(30,101,44,87)(31,86,45,100)(32,99,46,85)(33,112,47,98)(34,97,48,111)(35,110,49,96)(36,95,50,109)(37,108,51,94)(38,93,52,107)(39,106,53,92)(40,91,54,105)(41,104,55,90)(42,89,56,103)>;
G:=Group( (1,48,82,98,15,34,68,112)(2,49,83,99,16,35,69,85)(3,50,84,100,17,36,70,86)(4,51,57,101,18,37,71,87)(5,52,58,102,19,38,72,88)(6,53,59,103,20,39,73,89)(7,54,60,104,21,40,74,90)(8,55,61,105,22,41,75,91)(9,56,62,106,23,42,76,92)(10,29,63,107,24,43,77,93)(11,30,64,108,25,44,78,94)(12,31,65,109,26,45,79,95)(13,32,66,110,27,46,80,96)(14,33,67,111,28,47,81,97), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,81,15,67)(2,66,16,80)(3,79,17,65)(4,64,18,78)(5,77,19,63)(6,62,20,76)(7,75,21,61)(8,60,22,74)(9,73,23,59)(10,58,24,72)(11,71,25,57)(12,84,26,70)(13,69,27,83)(14,82,28,68)(29,88,43,102)(30,101,44,87)(31,86,45,100)(32,99,46,85)(33,112,47,98)(34,97,48,111)(35,110,49,96)(36,95,50,109)(37,108,51,94)(38,93,52,107)(39,106,53,92)(40,91,54,105)(41,104,55,90)(42,89,56,103) );
G=PermutationGroup([[(1,48,82,98,15,34,68,112),(2,49,83,99,16,35,69,85),(3,50,84,100,17,36,70,86),(4,51,57,101,18,37,71,87),(5,52,58,102,19,38,72,88),(6,53,59,103,20,39,73,89),(7,54,60,104,21,40,74,90),(8,55,61,105,22,41,75,91),(9,56,62,106,23,42,76,92),(10,29,63,107,24,43,77,93),(11,30,64,108,25,44,78,94),(12,31,65,109,26,45,79,95),(13,32,66,110,27,46,80,96),(14,33,67,111,28,47,81,97)], [(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(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)], [(1,81,15,67),(2,66,16,80),(3,79,17,65),(4,64,18,78),(5,77,19,63),(6,62,20,76),(7,75,21,61),(8,60,22,74),(9,73,23,59),(10,58,24,72),(11,71,25,57),(12,84,26,70),(13,69,27,83),(14,82,28,68),(29,88,43,102),(30,101,44,87),(31,86,45,100),(32,99,46,85),(33,112,47,98),(34,97,48,111),(35,110,49,96),(36,95,50,109),(37,108,51,94),(38,93,52,107),(39,106,53,92),(40,91,54,105),(41,104,55,90),(42,89,56,103)]])
82 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 | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| 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 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | C4×D7 | M4(2).8C22 | M4(2).31D14 |
| kernel | M4(2).31D14 | C28.46D4 | C4.12D28 | C2×C4.Dic7 | C14×M4(2) | C2×C4○D28 | C2×C4×D7 | C2×C7⋊D4 | C2×C28 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 4 | 3 | 6 | 3 | 6 | 12 | 12 | 6 | 2 | 12 |
Matrix representation of M4(2).31D14 ►in GL4(𝔽113) generated by
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 22 | 101 | 0 | 0 |
| 12 | 91 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 77 | 77 | 0 | 0 |
| 36 | 58 | 0 | 0 |
| 0 | 0 | 77 | 77 |
| 0 | 0 | 36 | 58 |
| 59 | 54 | 0 | 0 |
| 82 | 54 | 0 | 0 |
| 0 | 0 | 59 | 54 |
| 0 | 0 | 82 | 54 |
G:=sub<GL(4,GF(113))| [0,0,22,12,0,0,101,91,15,0,0,0,0,15,0,0],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[77,36,0,0,77,58,0,0,0,0,77,36,0,0,77,58],[59,82,0,0,54,54,0,0,0,0,59,82,0,0,54,54] >;
M4(2).31D14 in GAP, Magma, Sage, TeX
M_4(2)._{31}D_{14} % in TeX
G:=Group("M4(2).31D14"); // GroupNames label
G:=SmallGroup(448,666);
// by ID
G=gap.SmallGroup(448,666);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,422,58,1123,136,438,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^14=d^2=a^4,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^13>;
// generators/relations