metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).21D14, (C2×D28).4C4, (C4×D7).36D4, C4.151(D4×D7), C28.96(C2×D4), C4.10D4⋊6D7, (D7×M4(2))⋊7C2, (C2×C28).8C23, (C2×Q8).95D14, C28.46D4⋊7C2, C28.10D4⋊4C2, (Q8×C14).6C22, D14.3(C22⋊C4), (C2×D28).40C22, C4.Dic7.5C22, Dic7.17(C22⋊C4), (C7×M4(2)).13C22, C7⋊2(M4(2).8C22), (C2×C4×D7).3C4, (C2×C4).7(C4×D7), (C2×C28).7(C2×C4), (C2×C4×D7).4C22, C22.17(C2×C4×D7), C2.16(D7×C22⋊C4), (C2×C4).8(C22×D7), (C7×C4.10D4)⋊6C2, C14.15(C2×C22⋊C4), (C2×Q8⋊2D7).1C2, (C22×D7).3(C2×C4), (C2×C14).11(C22×C4), (C2×Dic7).86(C2×C4), SmallGroup(448,285)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).21D14
G = < a,b,c,d | a8=b2=d2=1, c14=a4, bab=a5, cac-1=ab, dad=a5b, bc=cb, bd=db, dcd=a4c13 >
Subgroups: 812 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C4.D4, C4.10D4, C4.10D4, C2×M4(2), C2×C4○D4, C7⋊C8, C56, C4×D7, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, M4(2).8C22, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, Q8⋊2D7, Q8×C14, C28.46D4, C28.10D4, C7×C4.10D4, D7×M4(2), C2×Q8⋊2D7, M4(2).21D14
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).21D14
►On 112 points
Generators in S112
(1 100 49 68 15 86 35 82)(2 87 36 69 16 101 50 83)(3 102 51 70 17 88 37 84)(4 89 38 71 18 103 52 57)(5 104 53 72 19 90 39 58)(6 91 40 73 20 105 54 59)(7 106 55 74 21 92 41 60)(8 93 42 75 22 107 56 61)(9 108 29 76 23 94 43 62)(10 95 44 77 24 109 30 63)(11 110 31 78 25 96 45 64)(12 97 46 79 26 111 32 65)(13 112 33 80 27 98 47 66)(14 99 48 81 28 85 34 67)
(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(48 56)(49 55)(50 54)(51 53)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(108 112)(109 111)
G:=sub<Sym(112)| (1,100,49,68,15,86,35,82)(2,87,36,69,16,101,50,83)(3,102,51,70,17,88,37,84)(4,89,38,71,18,103,52,57)(5,104,53,72,19,90,39,58)(6,91,40,73,20,105,54,59)(7,106,55,74,21,92,41,60)(8,93,42,75,22,107,56,61)(9,108,29,76,23,94,43,62)(10,95,44,77,24,109,30,63)(11,110,31,78,25,96,45,64)(12,97,46,79,26,111,32,65)(13,112,33,80,27,98,47,66)(14,99,48,81,28,85,34,67), (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(48,56)(49,55)(50,54)(51,53)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111)>;
G:=Group( (1,100,49,68,15,86,35,82)(2,87,36,69,16,101,50,83)(3,102,51,70,17,88,37,84)(4,89,38,71,18,103,52,57)(5,104,53,72,19,90,39,58)(6,91,40,73,20,105,54,59)(7,106,55,74,21,92,41,60)(8,93,42,75,22,107,56,61)(9,108,29,76,23,94,43,62)(10,95,44,77,24,109,30,63)(11,110,31,78,25,96,45,64)(12,97,46,79,26,111,32,65)(13,112,33,80,27,98,47,66)(14,99,48,81,28,85,34,67), (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(48,56)(49,55)(50,54)(51,53)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111) );
G=PermutationGroup([[(1,100,49,68,15,86,35,82),(2,87,36,69,16,101,50,83),(3,102,51,70,17,88,37,84),(4,89,38,71,18,103,52,57),(5,104,53,72,19,90,39,58),(6,91,40,73,20,105,54,59),(7,106,55,74,21,92,41,60),(8,93,42,75,22,107,56,61),(9,108,29,76,23,94,43,62),(10,95,44,77,24,109,30,63),(11,110,31,78,25,96,45,64),(12,97,46,79,26,111,32,65),(13,112,33,80,27,98,47,66),(14,99,48,81,28,85,34,67)], [(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,28),(23,27),(24,26),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(48,56),(49,55),(50,54),(51,53),(58,84),(59,83),(60,82),(61,81),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(85,107),(86,106),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(108,112),(109,111)]])
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 | 28A | ··· | 28F | 28G | ··· | 28L | 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 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 7 | 7 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 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).21D14 |
| kernel | M4(2).21D14 | C28.46D4 | C28.10D4 | C7×C4.10D4 | D7×M4(2) | C2×Q8⋊2D7 | C2×C4×D7 | C2×D28 | C4×D7 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | 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).21D14 ►in GL8(𝔽113)
| 0 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 | 15 | 83 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 0 | 1 |
| 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 15 | 98 | 0 | 112 |
| 25 | 0 | 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 88 | 0 | 88 | 0 | 0 | 0 | 0 |
| 88 | 0 | 79 | 0 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 | 15 | 83 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 98 |
| 88 | 0 | 88 | 0 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 25 | 0 | 0 | 0 | 0 |
| 34 | 0 | 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 79 | 0 | 88 | 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 | 15 | 0 | 1 | 112 |
G:=sub<GL(8,GF(113))| [0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,112,0,98,98,0,0,0,0,1,0,0,0,0,0,0,0,15,15,0,0,0,0,0,0,83,0,0,1],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,15,0,0,0,0,0,1,0,98,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[25,0,88,0,0,0,0,0,0,88,0,25,0,0,0,0,25,0,79,0,0,0,0,0,0,88,0,34,0,0,0,0,0,0,0,0,0,15,112,0,0,0,0,0,15,0,1,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,83,98],[88,0,34,0,0,0,0,0,0,25,0,79,0,0,0,0,88,0,25,0,0,0,0,0,0,25,0,88,0,0,0,0,0,0,0,0,1,0,0,15,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,112] >;
M4(2).21D14 in GAP, Magma, Sage, TeX
M_4(2)._{21}D_{14} % in TeX
G:=Group("M4(2).21D14"); // GroupNames label
G:=SmallGroup(448,285);
// by ID
G=gap.SmallGroup(448,285);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,232,219,58,570,136,438,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^14=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d=a^5*b,b*c=c*b,b*d=d*b,d*c*d=a^4*c^13>;
// generators/relations