metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.24D14, M4(2).22D14, C4≀C2⋊6D7, D4⋊D7⋊3C4, Q8⋊D7⋊3C4, C7⋊C8.34D4, D4.D7⋊3C4, D4.3(C4×D7), C7⋊Q16⋊3C4, Q8.3(C4×D7), C7⋊2(C8.26D4), D28.C4⋊8C2, D28.6(C2×C4), C4.202(D4×D7), C14.38(C4×D4), Dic14⋊C4⋊6C2, C4○D4.20D14, C28.361(C2×D4), Q8.Dic7⋊1C2, C28.53D4⋊5C2, C28.19(C22×C4), (C4×C28).50C22, Dic14.6(C2×C4), C42.D7⋊2C2, (C2×C28).263C23, D4.8D14.1C2, C4○D28.12C22, C4.Dic7.8C22, C22.8(D4⋊2D7), C2.22(Dic7⋊4D4), (C7×M4(2)).16C22, (C7×C4≀C2)⋊7C2, C7⋊C8.2(C2×C4), C4.19(C2×C4×D7), (C7×D4).6(C2×C4), (C7×Q8).6(C2×C4), (C2×C7⋊C8).50C22, (C7×C4○D4).4C22, (C2×C14).34(C4○D4), (C2×C4).369(C22×D7), SmallGroup(448,357)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).22D14
G = < a,b,c,d | a8=b2=c14=1, d2=a6b, bab=a5, cac-1=dad-1=a-1b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >
Subgroups: 412 in 104 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C42, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C8⋊C4, C4≀C2, C4≀C2, C8.C4, C8○D4, C4○D8, C7⋊C8, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8.26D4, C8×D7, C8⋊D7, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C4×C28, C7×M4(2), C4○D28, C7×C4○D4, C42.D7, Dic14⋊C4, C28.53D4, C7×C4≀C2, D28.C4, Q8.Dic7, D4.8D14, M4(2).22D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C22×D7, C8.26D4, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, M4(2).22D14
►On 112 points
(1 96 52 8 17 45 89 24)(2 25 53 97 18 9 90 46)(3 98 54 10 19 47 91 26)(4 27 55 85 20 11 92 48)(5 86 56 12 21 49 93 28)(6 15 43 87 22 13 94 50)(7 88 44 14 23 51 95 16)(29 108 101 76 83 68 61 36)(30 37 102 109 84 77 62 69)(31 110 103 78 71 70 63 38)(32 39 104 111 72 79 64 57)(33 112 105 80 73 58 65 40)(34 41 106 99 74 81 66 59)(35 100 107 82 75 60 67 42)
(1 17)(3 19)(5 21)(7 23)(9 25)(11 27)(13 15)(29 83)(31 71)(33 73)(35 75)(37 77)(39 79)(41 81)(44 95)(46 97)(48 85)(50 87)(52 89)(54 91)(56 93)(57 111)(59 99)(61 101)(63 103)(65 105)(67 107)(69 109)
(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 31 52 103 17 71 89 63)(2 62 90 84 18 102 53 30)(3 29 54 101 19 83 91 61)(4 60 92 82 20 100 55 42)(5 41 56 99 21 81 93 59)(6 58 94 80 22 112 43 40)(7 39 44 111 23 79 95 57)(8 70 96 78 24 110 45 38)(9 37 46 109 25 77 97 69)(10 68 98 76 26 108 47 36)(11 35 48 107 27 75 85 67)(12 66 86 74 28 106 49 34)(13 33 50 105 15 73 87 65)(14 64 88 72 16 104 51 32)
G:=sub<Sym(112)| (1,96,52,8,17,45,89,24)(2,25,53,97,18,9,90,46)(3,98,54,10,19,47,91,26)(4,27,55,85,20,11,92,48)(5,86,56,12,21,49,93,28)(6,15,43,87,22,13,94,50)(7,88,44,14,23,51,95,16)(29,108,101,76,83,68,61,36)(30,37,102,109,84,77,62,69)(31,110,103,78,71,70,63,38)(32,39,104,111,72,79,64,57)(33,112,105,80,73,58,65,40)(34,41,106,99,74,81,66,59)(35,100,107,82,75,60,67,42), (1,17)(3,19)(5,21)(7,23)(9,25)(11,27)(13,15)(29,83)(31,71)(33,73)(35,75)(37,77)(39,79)(41,81)(44,95)(46,97)(48,85)(50,87)(52,89)(54,91)(56,93)(57,111)(59,99)(61,101)(63,103)(65,105)(67,107)(69,109), (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), (1,31,52,103,17,71,89,63)(2,62,90,84,18,102,53,30)(3,29,54,101,19,83,91,61)(4,60,92,82,20,100,55,42)(5,41,56,99,21,81,93,59)(6,58,94,80,22,112,43,40)(7,39,44,111,23,79,95,57)(8,70,96,78,24,110,45,38)(9,37,46,109,25,77,97,69)(10,68,98,76,26,108,47,36)(11,35,48,107,27,75,85,67)(12,66,86,74,28,106,49,34)(13,33,50,105,15,73,87,65)(14,64,88,72,16,104,51,32)>;
G:=Group( (1,96,52,8,17,45,89,24)(2,25,53,97,18,9,90,46)(3,98,54,10,19,47,91,26)(4,27,55,85,20,11,92,48)(5,86,56,12,21,49,93,28)(6,15,43,87,22,13,94,50)(7,88,44,14,23,51,95,16)(29,108,101,76,83,68,61,36)(30,37,102,109,84,77,62,69)(31,110,103,78,71,70,63,38)(32,39,104,111,72,79,64,57)(33,112,105,80,73,58,65,40)(34,41,106,99,74,81,66,59)(35,100,107,82,75,60,67,42), (1,17)(3,19)(5,21)(7,23)(9,25)(11,27)(13,15)(29,83)(31,71)(33,73)(35,75)(37,77)(39,79)(41,81)(44,95)(46,97)(48,85)(50,87)(52,89)(54,91)(56,93)(57,111)(59,99)(61,101)(63,103)(65,105)(67,107)(69,109), (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), (1,31,52,103,17,71,89,63)(2,62,90,84,18,102,53,30)(3,29,54,101,19,83,91,61)(4,60,92,82,20,100,55,42)(5,41,56,99,21,81,93,59)(6,58,94,80,22,112,43,40)(7,39,44,111,23,79,95,57)(8,70,96,78,24,110,45,38)(9,37,46,109,25,77,97,69)(10,68,98,76,26,108,47,36)(11,35,48,107,27,75,85,67)(12,66,86,74,28,106,49,34)(13,33,50,105,15,73,87,65)(14,64,88,72,16,104,51,32) );
G=PermutationGroup([[(1,96,52,8,17,45,89,24),(2,25,53,97,18,9,90,46),(3,98,54,10,19,47,91,26),(4,27,55,85,20,11,92,48),(5,86,56,12,21,49,93,28),(6,15,43,87,22,13,94,50),(7,88,44,14,23,51,95,16),(29,108,101,76,83,68,61,36),(30,37,102,109,84,77,62,69),(31,110,103,78,71,70,63,38),(32,39,104,111,72,79,64,57),(33,112,105,80,73,58,65,40),(34,41,106,99,74,81,66,59),(35,100,107,82,75,60,67,42)], [(1,17),(3,19),(5,21),(7,23),(9,25),(11,27),(13,15),(29,83),(31,71),(33,73),(35,75),(37,77),(39,79),(41,81),(44,95),(46,97),(48,85),(50,87),(52,89),(54,91),(56,93),(57,111),(59,99),(61,101),(63,103),(65,105),(67,107),(69,109)], [(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)], [(1,31,52,103,17,71,89,63),(2,62,90,84,18,102,53,30),(3,29,54,101,19,83,91,61),(4,60,92,82,20,100,55,42),(5,41,56,99,21,81,93,59),(6,58,94,80,22,112,43,40),(7,39,44,111,23,79,95,57),(8,70,96,78,24,110,45,38),(9,37,46,109,25,77,97,69),(10,68,98,76,26,108,47,36),(11,35,48,107,27,75,85,67),(12,66,86,74,28,106,49,34),(13,33,50,105,15,73,87,65),(14,64,88,72,16,104,51,32)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28U | 28V | 28W | 28X | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 1 | 1 | 2 | 4 | 4 | 4 | 28 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4×D7 | C4×D7 | C8.26D4 | D4×D7 | D4⋊2D7 | M4(2).22D14 |
| kernel | M4(2).22D14 | C42.D7 | Dic14⋊C4 | C28.53D4 | C7×C4≀C2 | D28.C4 | Q8.Dic7 | D4.8D14 | D4⋊D7 | D4.D7 | Q8⋊D7 | C7⋊Q16 | C7⋊C8 | C4≀C2 | C2×C14 | C42 | M4(2) | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of M4(2).22D14 ►in GL4(𝔽113) generated by
| 0 | 112 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 15 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 105 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 106 |
| 0 | 0 | 106 | 0 |
| 0 | 99 | 0 | 0 |
| 14 | 0 | 0 | 0 |
G:=sub<GL(4,GF(113))| [0,98,0,0,112,0,0,0,0,0,0,15,0,0,1,0],[112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,112],[0,16,0,0,16,0,0,0,0,0,0,8,0,0,105,0],[0,0,0,14,0,0,99,0,0,106,0,0,106,0,0,0] >;
M4(2).22D14 in GAP, Magma, Sage, TeX
M_4(2)._{22}D_{14} % in TeX
G:=Group("M4(2).22D14"); // GroupNames label
G:=SmallGroup(448,357);
// by ID
G=gap.SmallGroup(448,357);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,219,58,136,1684,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^-1*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;
// generators/relations