metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.7Dic14, M4(2).1Dic7, C56.6(C2×C4), (C2×C8).76D14, C4.87(C2×D28), C28.22(C4⋊C4), (C2×C28).29Q8, C8.4(C2×Dic7), (C2×C28).169D4, C28.305(C2×D4), (C2×C4).150D28, C4.7(C4⋊Dic7), C56.C4⋊13C2, (C2×C56).62C22, (C2×C4).17Dic14, (C7×M4(2)).1C4, (C2×M4(2)).2D7, (C22×C14).17Q8, C7⋊3(M4(2).C4), (C2×C28).796C23, C28.174(C22×C4), (C22×C4).135D14, (C14×M4(2)).2C2, C22.7(C4⋊Dic7), C22.9(C2×Dic14), C4.28(C22×Dic7), C4.Dic7.36C22, (C22×C28).184C22, C14.53(C2×C4⋊C4), C2.15(C2×C4⋊Dic7), (C2×C14).41(C2×Q8), (C2×C14).17(C4⋊C4), (C2×C28).104(C2×C4), (C2×C4).22(C2×Dic7), (C2×C4).721(C22×D7), (C2×C4.Dic7).25C2, SmallGroup(448,659)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).Dic7
G = < a,b,c,d | a8=b2=1, c14=a4, d2=a4c7, bab=a5, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c13 >
Subgroups: 292 in 102 conjugacy classes, 67 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C8.C4, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×C28, C2×C28, C22×C14, M4(2).C4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C7×M4(2), C22×C28, C56.C4, C2×C4.Dic7, C14×M4(2), M4(2).Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic7, D14, C2×C4⋊C4, Dic14, D28, C2×Dic7, C22×D7, M4(2).C4, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C2×C4⋊Dic7, M4(2).Dic7
►On 112 points
(1 35 22 56 15 49 8 42)(2 36 23 29 16 50 9 43)(3 37 24 30 17 51 10 44)(4 38 25 31 18 52 11 45)(5 39 26 32 19 53 12 46)(6 40 27 33 20 54 13 47)(7 41 28 34 21 55 14 48)(57 85 64 92 71 99 78 106)(58 86 65 93 72 100 79 107)(59 87 66 94 73 101 80 108)(60 88 67 95 74 102 81 109)(61 89 68 96 75 103 82 110)(62 90 69 97 76 104 83 111)(63 91 70 98 77 105 84 112)
(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 57 22 78 15 71 8 64)(2 70 23 63 16 84 9 77)(3 83 24 76 17 69 10 62)(4 68 25 61 18 82 11 75)(5 81 26 74 19 67 12 60)(6 66 27 59 20 80 13 73)(7 79 28 72 21 65 14 58)(29 112 50 105 43 98 36 91)(30 97 51 90 44 111 37 104)(31 110 52 103 45 96 38 89)(32 95 53 88 46 109 39 102)(33 108 54 101 47 94 40 87)(34 93 55 86 48 107 41 100)(35 106 56 99 49 92 42 85)
G:=sub<Sym(112)| (1,35,22,56,15,49,8,42)(2,36,23,29,16,50,9,43)(3,37,24,30,17,51,10,44)(4,38,25,31,18,52,11,45)(5,39,26,32,19,53,12,46)(6,40,27,33,20,54,13,47)(7,41,28,34,21,55,14,48)(57,85,64,92,71,99,78,106)(58,86,65,93,72,100,79,107)(59,87,66,94,73,101,80,108)(60,88,67,95,74,102,81,109)(61,89,68,96,75,103,82,110)(62,90,69,97,76,104,83,111)(63,91,70,98,77,105,84,112), (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), (1,57,22,78,15,71,8,64)(2,70,23,63,16,84,9,77)(3,83,24,76,17,69,10,62)(4,68,25,61,18,82,11,75)(5,81,26,74,19,67,12,60)(6,66,27,59,20,80,13,73)(7,79,28,72,21,65,14,58)(29,112,50,105,43,98,36,91)(30,97,51,90,44,111,37,104)(31,110,52,103,45,96,38,89)(32,95,53,88,46,109,39,102)(33,108,54,101,47,94,40,87)(34,93,55,86,48,107,41,100)(35,106,56,99,49,92,42,85)>;
G:=Group( (1,35,22,56,15,49,8,42)(2,36,23,29,16,50,9,43)(3,37,24,30,17,51,10,44)(4,38,25,31,18,52,11,45)(5,39,26,32,19,53,12,46)(6,40,27,33,20,54,13,47)(7,41,28,34,21,55,14,48)(57,85,64,92,71,99,78,106)(58,86,65,93,72,100,79,107)(59,87,66,94,73,101,80,108)(60,88,67,95,74,102,81,109)(61,89,68,96,75,103,82,110)(62,90,69,97,76,104,83,111)(63,91,70,98,77,105,84,112), (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), (1,57,22,78,15,71,8,64)(2,70,23,63,16,84,9,77)(3,83,24,76,17,69,10,62)(4,68,25,61,18,82,11,75)(5,81,26,74,19,67,12,60)(6,66,27,59,20,80,13,73)(7,79,28,72,21,65,14,58)(29,112,50,105,43,98,36,91)(30,97,51,90,44,111,37,104)(31,110,52,103,45,96,38,89)(32,95,53,88,46,109,39,102)(33,108,54,101,47,94,40,87)(34,93,55,86,48,107,41,100)(35,106,56,99,49,92,42,85) );
G=PermutationGroup([[(1,35,22,56,15,49,8,42),(2,36,23,29,16,50,9,43),(3,37,24,30,17,51,10,44),(4,38,25,31,18,52,11,45),(5,39,26,32,19,53,12,46),(6,40,27,33,20,54,13,47),(7,41,28,34,21,55,14,48),(57,85,64,92,71,99,78,106),(58,86,65,93,72,100,79,107),(59,87,66,94,73,101,80,108),(60,88,67,95,74,102,81,109),(61,89,68,96,75,103,82,110),(62,90,69,97,76,104,83,111),(63,91,70,98,77,105,84,112)], [(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)], [(1,57,22,78,15,71,8,64),(2,70,23,63,16,84,9,77),(3,83,24,76,17,69,10,62),(4,68,25,61,18,82,11,75),(5,81,26,74,19,67,12,60),(6,66,27,59,20,80,13,73),(7,79,28,72,21,65,14,58),(29,112,50,105,43,98,36,91),(30,97,51,90,44,111,37,104),(31,110,52,103,45,96,38,89),(32,95,53,88,46,109,39,102),(33,108,54,101,47,94,40,87),(34,93,55,86,48,107,41,100),(35,106,56,99,49,92,42,85)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | - | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D7 | D14 | Dic7 | D14 | Dic14 | D28 | Dic14 | M4(2).C4 | M4(2).Dic7 |
| kernel | M4(2).Dic7 | C56.C4 | C2×C4.Dic7 | C14×M4(2) | C7×M4(2) | C2×C28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 1 | 1 | 3 | 6 | 12 | 3 | 6 | 12 | 6 | 2 | 12 |
Matrix representation of M4(2).Dic7 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
G:=sub<GL(4,GF(113))| [0,98,0,0,1,0,0,0,0,0,0,1,0,0,15,0],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112],[60,0,0,0,0,60,0,0,0,0,32,0,0,0,0,32],[0,0,98,0,0,0,0,98,1,0,0,0,0,1,0,0] >;
M4(2).Dic7 in GAP, Magma, Sage, TeX
M_4(2).{\rm Dic}_7 % in TeX
G:=Group("M4(2).Dic7"); // GroupNames label
G:=SmallGroup(448,659);
// by ID
G=gap.SmallGroup(448,659);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,387,100,136,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^14=a^4,d^2=a^4*c^7,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^13>;
// generators/relations