metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)⋊4Dic7, C28.13(C4⋊C4), (C2×C28).13Q8, (C2×C28).112D4, (C2×C4).130D28, (C7×M4(2))⋊7C4, (C2×C14).4C42, C4.4(C4⋊Dic7), (C2×C4).7Dic14, C23.10(C4×D7), C4.51(D14⋊C4), C23.D7.2C4, (C22×C4).64D14, C22.4(C4×Dic7), C28.88(C22⋊C4), C7⋊3(M4(2)⋊4C4), C4.13(Dic7⋊C4), (C2×M4(2)).10D7, C4.19(C23.D7), C22.21(D14⋊C4), (C14×M4(2)).14C2, C22.14(Dic7⋊C4), (C22×C28).128C22, C14.18(C2.C42), C2.18(C14.C42), C23.21D14.11C2, (C2×C7⋊C8)⋊3C4, (C2×C28).65(C2×C4), (C2×C4).142(C4×D7), (C2×C14).41(C4⋊C4), (C2×C4).15(C2×Dic7), (C2×C4).235(C7⋊D4), (C22×C14).33(C2×C4), (C2×C4.Dic7).13C2, (C2×C14).12(C22⋊C4), SmallGroup(448,116)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊4Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=cac-1=a5, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=c-1 >
Subgroups: 324 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, Dic7, C28, C2×C14, C2×C14, C42⋊C2, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×C28, C22×C14, M4(2)⋊4C4, C2×C7⋊C8, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C2×C4.Dic7, C23.21D14, C14×M4(2), M4(2)⋊4Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, M4(2)⋊4C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, M4(2)⋊4Dic7
►On 112 points
Generators in S112
(1 102 29 81 11 109 38 74)(2 110 30 75 12 103 39 82)(3 104 31 83 13 111 40 76)(4 112 32 77 14 105 41 84)(5 106 33 71 8 99 42 78)(6 100 34 79 9 107 36 72)(7 108 35 73 10 101 37 80)(15 93 43 59 22 86 50 66)(16 87 44 67 23 94 51 60)(17 95 45 61 24 88 52 68)(18 89 46 69 25 96 53 62)(19 97 47 63 26 90 54 70)(20 91 48 57 27 98 55 64)(21 85 49 65 28 92 56 58)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 26)(9 27)(10 28)(11 22)(12 23)(13 24)(14 25)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 55)(37 56)(38 50)(39 51)(40 52)(41 53)(42 54)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 71)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)(97 99)(98 100)
(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 29)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 42)(15 50)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 43)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(57 104 64 111)(58 103 65 110)(59 102 66 109)(60 101 67 108)(61 100 68 107)(62 99 69 106)(63 112 70 105)(71 96 78 89)(72 95 79 88)(73 94 80 87)(74 93 81 86)(75 92 82 85)(76 91 83 98)(77 90 84 97)
G:=sub<Sym(112)| (1,102,29,81,11,109,38,74)(2,110,30,75,12,103,39,82)(3,104,31,83,13,111,40,76)(4,112,32,77,14,105,41,84)(5,106,33,71,8,99,42,78)(6,100,34,79,9,107,36,72)(7,108,35,73,10,101,37,80)(15,93,43,59,22,86,50,66)(16,87,44,67,23,94,51,60)(17,95,45,61,24,88,52,68)(18,89,46,69,25,96,53,62)(19,97,47,63,26,90,54,70)(20,91,48,57,27,98,55,64)(21,85,49,65,28,92,56,58), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,26)(9,27)(10,28)(11,22)(12,23)(13,24)(14,25)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,55)(37,56)(38,50)(39,51)(40,52)(41,53)(42,54)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,71)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,99)(98,100), (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), (1,29)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,42)(15,50)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(57,104,64,111)(58,103,65,110)(59,102,66,109)(60,101,67,108)(61,100,68,107)(62,99,69,106)(63,112,70,105)(71,96,78,89)(72,95,79,88)(73,94,80,87)(74,93,81,86)(75,92,82,85)(76,91,83,98)(77,90,84,97)>;
G:=Group( (1,102,29,81,11,109,38,74)(2,110,30,75,12,103,39,82)(3,104,31,83,13,111,40,76)(4,112,32,77,14,105,41,84)(5,106,33,71,8,99,42,78)(6,100,34,79,9,107,36,72)(7,108,35,73,10,101,37,80)(15,93,43,59,22,86,50,66)(16,87,44,67,23,94,51,60)(17,95,45,61,24,88,52,68)(18,89,46,69,25,96,53,62)(19,97,47,63,26,90,54,70)(20,91,48,57,27,98,55,64)(21,85,49,65,28,92,56,58), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,26)(9,27)(10,28)(11,22)(12,23)(13,24)(14,25)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,55)(37,56)(38,50)(39,51)(40,52)(41,53)(42,54)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,71)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,99)(98,100), (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), (1,29)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,42)(15,50)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(57,104,64,111)(58,103,65,110)(59,102,66,109)(60,101,67,108)(61,100,68,107)(62,99,69,106)(63,112,70,105)(71,96,78,89)(72,95,79,88)(73,94,80,87)(74,93,81,86)(75,92,82,85)(76,91,83,98)(77,90,84,97) );
G=PermutationGroup([[(1,102,29,81,11,109,38,74),(2,110,30,75,12,103,39,82),(3,104,31,83,13,111,40,76),(4,112,32,77,14,105,41,84),(5,106,33,71,8,99,42,78),(6,100,34,79,9,107,36,72),(7,108,35,73,10,101,37,80),(15,93,43,59,22,86,50,66),(16,87,44,67,23,94,51,60),(17,95,45,61,24,88,52,68),(18,89,46,69,25,96,53,62),(19,97,47,63,26,90,54,70),(20,91,48,57,27,98,55,64),(21,85,49,65,28,92,56,58)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,26),(9,27),(10,28),(11,22),(12,23),(13,24),(14,25),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,55),(37,56),(38,50),(39,51),(40,52),(41,53),(42,54),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,71),(85,101),(86,102),(87,103),(88,104),(89,105),(90,106),(91,107),(92,108),(93,109),(94,110),(95,111),(96,112),(97,99),(98,100)], [(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)], [(1,29),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,42),(15,50),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,43),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(57,104,64,111),(58,103,65,110),(59,102,66,109),(60,101,67,108),(61,100,68,107),(62,99,69,106),(63,112,70,105),(71,96,78,89),(72,95,79,88),(73,94,80,87),(74,93,81,86),(75,92,82,85),(76,91,83,98),(77,90,84,97)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D7 | Dic7 | D14 | Dic14 | C4×D7 | D28 | C7⋊D4 | C4×D7 | M4(2)⋊4C4 | M4(2)⋊4Dic7 |
| kernel | M4(2)⋊4Dic7 | C2×C4.Dic7 | C23.21D14 | C14×M4(2) | C2×C7⋊C8 | C23.D7 | C7×M4(2) | C2×C28 | C2×C28 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 3 | 1 | 3 | 6 | 3 | 6 | 6 | 6 | 12 | 6 | 2 | 12 |
Matrix representation of M4(2)⋊4Dic7 ►in GL4(𝔽113) generated by
| 22 | 101 | 69 | 24 |
| 12 | 91 | 89 | 44 |
| 60 | 107 | 91 | 12 |
| 6 | 38 | 101 | 22 |
| 22 | 101 | 0 | 0 |
| 12 | 91 | 0 | 0 |
| 22 | 101 | 91 | 12 |
| 12 | 91 | 101 | 22 |
| 0 | 1 | 0 | 0 |
| 112 | 79 | 0 | 0 |
| 0 | 1 | 0 | 112 |
| 112 | 79 | 1 | 34 |
| 9 | 67 | 0 | 0 |
| 100 | 104 | 0 | 0 |
| 12 | 90 | 98 | 0 |
| 21 | 101 | 58 | 15 |
G:=sub<GL(4,GF(113))| [22,12,60,6,101,91,107,38,69,89,91,101,24,44,12,22],[22,12,22,12,101,91,101,91,0,0,91,101,0,0,12,22],[0,112,0,112,1,79,1,79,0,0,0,1,0,0,112,34],[9,100,12,21,67,104,90,101,0,0,98,58,0,0,0,15] >;
M4(2)⋊4Dic7 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_4{\rm Dic}_7 % in TeX
G:=Group("M4(2):4Dic7"); // GroupNames label
G:=SmallGroup(448,116);
// by ID
G=gap.SmallGroup(448,116);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,184,1123,136,851,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=c^7,b*a*b=c*a*c^-1=a^5,d*a*d^-1=a^5*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations