metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.46D4, C8.25D28, M5(2)⋊3D7, C28.6M4(2), (C2×Dic7).C8, (C22×D7).C8, C7⋊1(C23.C8), C22.5(C8×D7), (C2×C8).152D14, C8.46(C7⋊D4), (C7×M5(2))⋊7C2, C28.C8⋊10C2, C2.10(D14⋊C8), C4.42(D14⋊C4), C4.10(C8⋊D7), C14.9(C22⋊C8), C28.57(C22⋊C4), (C2×C56).220C22, (C2×C7⋊C8).2C4, (C2×C4×D7).1C4, (C2×C14).3(C2×C8), (C2×C28).52(C2×C4), (C2×C4).137(C4×D7), (C2×C8⋊D7).10C2, SmallGroup(448,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M5(2)⋊D7
G = < a,b,c,d | a16=b2=c7=d2=1, bab=a9, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >
Subgroups: 276 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C16, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, D14, C2×C14, M5(2), M5(2), C2×M4(2), C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C23.C8, C7⋊C16, C112, C8⋊D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C28.C8, C7×M5(2), C2×C8⋊D7, M5(2)⋊D7
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), D14, C22⋊C8, C4×D7, D28, C7⋊D4, C23.C8, C8×D7, C8⋊D7, D14⋊C4, D14⋊C8, M5(2)⋊D7
►On 112 points
Generators in S112
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(98 106)(100 108)(102 110)(104 112)
(1 111 47 74 23 96 58)(2 112 48 75 24 81 59)(3 97 33 76 25 82 60)(4 98 34 77 26 83 61)(5 99 35 78 27 84 62)(6 100 36 79 28 85 63)(7 101 37 80 29 86 64)(8 102 38 65 30 87 49)(9 103 39 66 31 88 50)(10 104 40 67 32 89 51)(11 105 41 68 17 90 52)(12 106 42 69 18 91 53)(13 107 43 70 19 92 54)(14 108 44 71 20 93 55)(15 109 45 72 21 94 56)(16 110 46 73 22 95 57)
(1 58)(2 51)(3 52)(4 61)(5 62)(6 55)(7 56)(8 49)(9 50)(10 59)(11 60)(12 53)(13 54)(14 63)(15 64)(16 57)(17 33)(18 42)(19 43)(20 36)(21 37)(22 46)(23 47)(24 40)(25 41)(26 34)(27 35)(28 44)(29 45)(30 38)(31 39)(32 48)(67 75)(68 76)(71 79)(72 80)(81 104)(82 105)(83 98)(84 99)(85 108)(86 109)(87 102)(88 103)(89 112)(90 97)(91 106)(92 107)(93 100)(94 101)(95 110)(96 111)
G:=sub<Sym(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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(98,106)(100,108)(102,110)(104,112), (1,111,47,74,23,96,58)(2,112,48,75,24,81,59)(3,97,33,76,25,82,60)(4,98,34,77,26,83,61)(5,99,35,78,27,84,62)(6,100,36,79,28,85,63)(7,101,37,80,29,86,64)(8,102,38,65,30,87,49)(9,103,39,66,31,88,50)(10,104,40,67,32,89,51)(11,105,41,68,17,90,52)(12,106,42,69,18,91,53)(13,107,43,70,19,92,54)(14,108,44,71,20,93,55)(15,109,45,72,21,94,56)(16,110,46,73,22,95,57), (1,58)(2,51)(3,52)(4,61)(5,62)(6,55)(7,56)(8,49)(9,50)(10,59)(11,60)(12,53)(13,54)(14,63)(15,64)(16,57)(17,33)(18,42)(19,43)(20,36)(21,37)(22,46)(23,47)(24,40)(25,41)(26,34)(27,35)(28,44)(29,45)(30,38)(31,39)(32,48)(67,75)(68,76)(71,79)(72,80)(81,104)(82,105)(83,98)(84,99)(85,108)(86,109)(87,102)(88,103)(89,112)(90,97)(91,106)(92,107)(93,100)(94,101)(95,110)(96,111)>;
G:=Group( (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(65,73)(67,75)(69,77)(71,79)(81,89)(83,91)(85,93)(87,95)(98,106)(100,108)(102,110)(104,112), (1,111,47,74,23,96,58)(2,112,48,75,24,81,59)(3,97,33,76,25,82,60)(4,98,34,77,26,83,61)(5,99,35,78,27,84,62)(6,100,36,79,28,85,63)(7,101,37,80,29,86,64)(8,102,38,65,30,87,49)(9,103,39,66,31,88,50)(10,104,40,67,32,89,51)(11,105,41,68,17,90,52)(12,106,42,69,18,91,53)(13,107,43,70,19,92,54)(14,108,44,71,20,93,55)(15,109,45,72,21,94,56)(16,110,46,73,22,95,57), (1,58)(2,51)(3,52)(4,61)(5,62)(6,55)(7,56)(8,49)(9,50)(10,59)(11,60)(12,53)(13,54)(14,63)(15,64)(16,57)(17,33)(18,42)(19,43)(20,36)(21,37)(22,46)(23,47)(24,40)(25,41)(26,34)(27,35)(28,44)(29,45)(30,38)(31,39)(32,48)(67,75)(68,76)(71,79)(72,80)(81,104)(82,105)(83,98)(84,99)(85,108)(86,109)(87,102)(88,103)(89,112)(90,97)(91,106)(92,107)(93,100)(94,101)(95,110)(96,111) );
G=PermutationGroup([[(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63),(65,73),(67,75),(69,77),(71,79),(81,89),(83,91),(85,93),(87,95),(98,106),(100,108),(102,110),(104,112)], [(1,111,47,74,23,96,58),(2,112,48,75,24,81,59),(3,97,33,76,25,82,60),(4,98,34,77,26,83,61),(5,99,35,78,27,84,62),(6,100,36,79,28,85,63),(7,101,37,80,29,86,64),(8,102,38,65,30,87,49),(9,103,39,66,31,88,50),(10,104,40,67,32,89,51),(11,105,41,68,17,90,52),(12,106,42,69,18,91,53),(13,107,43,70,19,92,54),(14,108,44,71,20,93,55),(15,109,45,72,21,94,56),(16,110,46,73,22,95,57)], [(1,58),(2,51),(3,52),(4,61),(5,62),(6,55),(7,56),(8,49),(9,50),(10,59),(11,60),(12,53),(13,54),(14,63),(15,64),(16,57),(17,33),(18,42),(19,43),(20,36),(21,37),(22,46),(23,47),(24,40),(25,41),(26,34),(27,35),(28,44),(29,45),(30,38),(31,39),(32,48),(67,75),(68,76),(71,79),(72,80),(81,104),(82,105),(83,98),(84,99),(85,108),(86,109),(87,102),(88,103),(89,112),(90,97),(91,106),(92,107),(93,100),(94,101),(95,110),(96,111)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 28 | 1 | 1 | 2 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 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 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D7 | M4(2) | D14 | D28 | C7⋊D4 | C4×D7 | C8⋊D7 | C8×D7 | C23.C8 | M5(2)⋊D7 |
| kernel | M5(2)⋊D7 | C28.C8 | C7×M5(2) | C2×C8⋊D7 | C2×C7⋊C8 | C2×C4×D7 | C2×Dic7 | C22×D7 | C56 | M5(2) | C28 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 3 | 2 | 3 | 6 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of M5(2)⋊D7 ►in GL4(𝔽113) generated by
| 98 | 0 | 80 | 71 |
| 0 | 98 | 42 | 0 |
| 97 | 107 | 15 | 0 |
| 98 | 21 | 0 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 33 | 112 | 0 |
| 80 | 79 | 0 | 112 |
| 0 | 1 | 0 | 0 |
| 112 | 9 | 0 | 0 |
| 0 | 0 | 34 | 1 |
| 0 | 0 | 53 | 88 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 88 | 112 |
| 0 | 0 | 59 | 25 |
G:=sub<GL(4,GF(113))| [98,0,97,98,0,98,107,21,80,42,15,0,71,0,0,15],[1,0,0,80,0,1,33,79,0,0,112,0,0,0,0,112],[0,112,0,0,1,9,0,0,0,0,34,53,0,0,1,88],[0,1,0,0,1,0,0,0,0,0,88,59,0,0,112,25] >;
M5(2)⋊D7 in GAP, Magma, Sage, TeX
M_5(2)\rtimes D_7
% in TeX
G:=Group("M5(2):D7"); // GroupNames label
G:=SmallGroup(448,71);
// by ID
G=gap.SmallGroup(448,71);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,100,570,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^7=d^2=1,b*a*b=a^9,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations