metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊6M4(2), (C2×C8)⋊20D14, D14⋊C8⋊37C2, C4.88(C2×D28), (C2×C56)⋊35C22, (C2×C28).170D4, C28.306(C2×D4), (C2×C4).151D28, (C2×M4(2))⋊6D7, (C23×D7).7C4, C23.54(C4×D7), C4.38(D14⋊C4), C7⋊2(C24.4C4), C2.20(D7×M4(2)), C28.25(C22⋊C4), (C14×M4(2))⋊16C2, (C2×C28).866C23, (C22×C4).349D14, C14.31(C2×M4(2)), C22.27(D14⋊C4), (C22×Dic7).16C4, (C22×C28).185C22, (C2×C4×D7).9C4, (C2×C7⋊C8)⋊28C22, (D7×C22×C4).2C2, C2.27(C2×D14⋊C4), (C2×C4).159(C4×D7), C4.132(C2×C7⋊D4), C22.145(C2×C4×D7), (C2×C28).105(C2×C4), C14.55(C2×C22⋊C4), (C2×C4.Dic7)⋊14C2, (C2×C4×D7).284C22, (C2×C4).195(C7⋊D4), (C22×C14).66(C2×C4), (C22×D7).63(C2×C4), (C2×C4).808(C22×D7), (C2×C14).19(C22⋊C4), (C2×C14).136(C22×C4), (C2×Dic7).102(C2×C4), SmallGroup(448,660)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14⋊6M4(2)
G = < a,b,c,d | a14=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=c5 >
Subgroups: 932 in 190 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C23, D7, C14, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, C2×M4(2), C2×M4(2), C23×C4, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C24.4C4, C2×C7⋊C8, C4.Dic7, C2×C56, C7×M4(2), C2×C4×D7, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, D14⋊C8, C2×C4.Dic7, C14×M4(2), D7×C22×C4, D14⋊6M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, M4(2), C22×C4, C2×D4, D14, C2×C22⋊C4, C2×M4(2), C4×D7, D28, C7⋊D4, C22×D7, C24.4C4, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, D7×M4(2), C2×D14⋊C4, D14⋊6M4(2)
►On 112 points
(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 7)(2 6)(3 5)(8 14)(9 13)(10 12)(15 25)(16 24)(17 23)(18 22)(19 21)(26 28)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(71 80)(72 79)(73 78)(74 77)(75 76)(81 84)(82 83)(85 87)(88 98)(89 97)(90 96)(91 95)(92 94)(99 100)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)
(1 107 17 64 90 76 53 36)(2 108 18 65 91 77 54 37)(3 109 19 66 92 78 55 38)(4 110 20 67 93 79 56 39)(5 111 21 68 94 80 43 40)(6 112 22 69 95 81 44 41)(7 99 23 70 96 82 45 42)(8 100 24 57 97 83 46 29)(9 101 25 58 98 84 47 30)(10 102 26 59 85 71 48 31)(11 103 27 60 86 72 49 32)(12 104 28 61 87 73 50 33)(13 105 15 62 88 74 51 34)(14 106 16 63 89 75 52 35)
(1 97)(2 98)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 43)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
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), (1,7)(2,6)(3,5)(8,14)(9,13)(10,12)(15,25)(16,24)(17,23)(18,22)(19,21)(26,28)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(71,80)(72,79)(73,78)(74,77)(75,76)(81,84)(82,83)(85,87)(88,98)(89,97)(90,96)(91,95)(92,94)(99,100)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107), (1,107,17,64,90,76,53,36)(2,108,18,65,91,77,54,37)(3,109,19,66,92,78,55,38)(4,110,20,67,93,79,56,39)(5,111,21,68,94,80,43,40)(6,112,22,69,95,81,44,41)(7,99,23,70,96,82,45,42)(8,100,24,57,97,83,46,29)(9,101,25,58,98,84,47,30)(10,102,26,59,85,71,48,31)(11,103,27,60,86,72,49,32)(12,104,28,61,87,73,50,33)(13,105,15,62,88,74,51,34)(14,106,16,63,89,75,52,35), (1,97)(2,98)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,43)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)>;
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), (1,7)(2,6)(3,5)(8,14)(9,13)(10,12)(15,25)(16,24)(17,23)(18,22)(19,21)(26,28)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(71,80)(72,79)(73,78)(74,77)(75,76)(81,84)(82,83)(85,87)(88,98)(89,97)(90,96)(91,95)(92,94)(99,100)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107), (1,107,17,64,90,76,53,36)(2,108,18,65,91,77,54,37)(3,109,19,66,92,78,55,38)(4,110,20,67,93,79,56,39)(5,111,21,68,94,80,43,40)(6,112,22,69,95,81,44,41)(7,99,23,70,96,82,45,42)(8,100,24,57,97,83,46,29)(9,101,25,58,98,84,47,30)(10,102,26,59,85,71,48,31)(11,103,27,60,86,72,49,32)(12,104,28,61,87,73,50,33)(13,105,15,62,88,74,51,34)(14,106,16,63,89,75,52,35), (1,97)(2,98)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,43)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112) );
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)], [(1,7),(2,6),(3,5),(8,14),(9,13),(10,12),(15,25),(16,24),(17,23),(18,22),(19,21),(26,28),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(71,80),(72,79),(73,78),(74,77),(75,76),(81,84),(82,83),(85,87),(88,98),(89,97),(90,96),(91,95),(92,94),(99,100),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107)], [(1,107,17,64,90,76,53,36),(2,108,18,65,91,77,54,37),(3,109,19,66,92,78,55,38),(4,110,20,67,93,79,56,39),(5,111,21,68,94,80,43,40),(6,112,22,69,95,81,44,41),(7,99,23,70,96,82,45,42),(8,100,24,57,97,83,46,29),(9,101,25,58,98,84,47,30),(10,102,26,59,85,71,48,31),(11,103,27,60,86,72,49,32),(12,104,28,61,87,73,50,33),(13,105,15,62,88,74,51,34),(14,106,16,63,89,75,52,35)], [(1,97),(2,98),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,43),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 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 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | M4(2) | D14 | D14 | C4×D7 | D28 | C7⋊D4 | C4×D7 | D7×M4(2) |
| kernel | D14⋊6M4(2) | D14⋊C8 | C2×C4.Dic7 | C14×M4(2) | D7×C22×C4 | C2×C4×D7 | C22×Dic7 | C23×D7 | C2×C28 | C2×M4(2) | D14 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 3 | 8 | 6 | 3 | 6 | 12 | 12 | 6 | 12 |
Matrix representation of D14⋊6M4(2) ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 80 |
| 0 | 0 | 59 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 112 |
| 0 | 0 | 59 | 88 |
| 0 | 112 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 104 | 75 |
| 0 | 0 | 20 | 9 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,1,59,0,0,80,88],[1,0,0,0,0,1,0,0,0,0,25,59,0,0,112,88],[0,98,0,0,112,0,0,0,0,0,104,20,0,0,75,9],[112,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112] >;
D14⋊6M4(2) in GAP, Magma, Sage, TeX
D_{14}\rtimes_6M_4(2) % in TeX
G:=Group("D14:6M4(2)"); // GroupNames label
G:=SmallGroup(448,660);
// by ID
G=gap.SmallGroup(448,660);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,387,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^5>;
// generators/relations