metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊6SD16, D28.16D4, (C2×C8)⋊17D14, (C2×Q8)⋊3D14, (C7×D4).9D4, C4.62(D4×D7), D14⋊C8⋊32C2, C28.47(C2×D4), (C2×C56)⋊33C22, D14⋊3Q8⋊3C2, (C2×SD16)⋊10D7, C7⋊4(C22⋊SD16), D4.8(C7⋊D4), C2.D56⋊35C2, C2.28(D7×SD16), (Q8×C14)⋊3C22, C14.57C22≀C2, (C14×SD16)⋊20C2, (C2×D4).146D14, D4⋊Dic7⋊33C2, C4⋊Dic7⋊20C22, (C2×Dic7).71D4, C14.45(C2×SD16), (C22×D7).91D4, C22.266(D4×D7), C2.28(D56⋊C2), C14.78(C8⋊C22), (C2×C28).446C23, (D4×C14).95C22, C2.25(C23⋊D14), (C2×D28).120C22, (C2×D4×D7).6C2, (C2×C7⋊C8)⋊8C22, (C2×Q8⋊D7)⋊17C2, C4.42(C2×C7⋊D4), (C2×C4×D7).47C22, (C2×C14).358(C2×D4), (C2×C4).535(C22×D7), SmallGroup(448,703)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D14⋊6SD16
G = < a,b,c,d | a14=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=c3 >
Subgroups: 1284 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C2×SD16, C22×D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C22⋊SD16, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, D14⋊C4, Q8⋊D7, C2×C56, C7×SD16, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, D14⋊C8, C2.D56, D4⋊Dic7, C2×Q8⋊D7, D14⋊3Q8, C14×SD16, C2×D4×D7, D14⋊6SD16
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8⋊C22, C7⋊D4, C22×D7, C22⋊SD16, D4×D7, C2×C7⋊D4, D7×SD16, D56⋊C2, C23⋊D14, D14⋊6SD16
►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)
(1 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 33)(30 32)(34 42)(35 41)(36 40)(37 39)(43 50)(44 49)(45 48)(46 47)(51 56)(52 55)(53 54)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 72)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 94)(86 93)(87 92)(88 91)(89 90)(95 98)(96 97)(99 107)(100 106)(101 105)(102 104)(108 112)(109 111)
(1 107 47 15 79 64 90 35)(2 108 48 16 80 65 91 36)(3 109 49 17 81 66 92 37)(4 110 50 18 82 67 93 38)(5 111 51 19 83 68 94 39)(6 112 52 20 84 69 95 40)(7 99 53 21 71 70 96 41)(8 100 54 22 72 57 97 42)(9 101 55 23 73 58 98 29)(10 102 56 24 74 59 85 30)(11 103 43 25 75 60 86 31)(12 104 44 26 76 61 87 32)(13 105 45 27 77 62 88 33)(14 106 46 28 78 63 89 34)
(1 54)(2 55)(3 56)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 85)(82 86)(83 87)(84 88)
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,21)(16,20)(17,19)(22,28)(23,27)(24,26)(29,33)(30,32)(34,42)(35,41)(36,40)(37,39)(43,50)(44,49)(45,48)(46,47)(51,56)(52,55)(53,54)(57,63)(58,62)(59,61)(64,70)(65,69)(66,68)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,94)(86,93)(87,92)(88,91)(89,90)(95,98)(96,97)(99,107)(100,106)(101,105)(102,104)(108,112)(109,111), (1,107,47,15,79,64,90,35)(2,108,48,16,80,65,91,36)(3,109,49,17,81,66,92,37)(4,110,50,18,82,67,93,38)(5,111,51,19,83,68,94,39)(6,112,52,20,84,69,95,40)(7,99,53,21,71,70,96,41)(8,100,54,22,72,57,97,42)(9,101,55,23,73,58,98,29)(10,102,56,24,74,59,85,30)(11,103,43,25,75,60,86,31)(12,104,44,26,76,61,87,32)(13,105,45,27,77,62,88,33)(14,106,46,28,78,63,89,34), (1,54)(2,55)(3,56)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,85)(82,86)(83,87)(84,88)>;
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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,21)(16,20)(17,19)(22,28)(23,27)(24,26)(29,33)(30,32)(34,42)(35,41)(36,40)(37,39)(43,50)(44,49)(45,48)(46,47)(51,56)(52,55)(53,54)(57,63)(58,62)(59,61)(64,70)(65,69)(66,68)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,94)(86,93)(87,92)(88,91)(89,90)(95,98)(96,97)(99,107)(100,106)(101,105)(102,104)(108,112)(109,111), (1,107,47,15,79,64,90,35)(2,108,48,16,80,65,91,36)(3,109,49,17,81,66,92,37)(4,110,50,18,82,67,93,38)(5,111,51,19,83,68,94,39)(6,112,52,20,84,69,95,40)(7,99,53,21,71,70,96,41)(8,100,54,22,72,57,97,42)(9,101,55,23,73,58,98,29)(10,102,56,24,74,59,85,30)(11,103,43,25,75,60,86,31)(12,104,44,26,76,61,87,32)(13,105,45,27,77,62,88,33)(14,106,46,28,78,63,89,34), (1,54)(2,55)(3,56)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,85)(82,86)(83,87)(84,88) );
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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,21),(16,20),(17,19),(22,28),(23,27),(24,26),(29,33),(30,32),(34,42),(35,41),(36,40),(37,39),(43,50),(44,49),(45,48),(46,47),(51,56),(52,55),(53,54),(57,63),(58,62),(59,61),(64,70),(65,69),(66,68),(71,72),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,94),(86,93),(87,92),(88,91),(89,90),(95,98),(96,97),(99,107),(100,106),(101,105),(102,104),(108,112),(109,111)], [(1,107,47,15,79,64,90,35),(2,108,48,16,80,65,91,36),(3,109,49,17,81,66,92,37),(4,110,50,18,82,67,93,38),(5,111,51,19,83,68,94,39),(6,112,52,20,84,69,95,40),(7,99,53,21,71,70,96,41),(8,100,54,22,72,57,97,42),(9,101,55,23,73,58,98,29),(10,102,56,24,74,59,85,30),(11,103,43,25,75,60,86,31),(12,104,44,26,76,61,87,32),(13,105,45,27,77,62,88,33),(14,106,46,28,78,63,89,34)], [(1,54),(2,55),(3,56),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,89),(72,90),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,85),(82,86),(83,87),(84,88)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 8 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | SD16 | D14 | D14 | D14 | C7⋊D4 | C8⋊C22 | D4×D7 | D4×D7 | D7×SD16 | D56⋊C2 |
| kernel | D14⋊6SD16 | D14⋊C8 | C2.D56 | D4⋊Dic7 | C2×Q8⋊D7 | D14⋊3Q8 | C14×SD16 | C2×D4×D7 | D28 | C2×Dic7 | C7×D4 | C22×D7 | C2×SD16 | D14 | C2×C8 | C2×D4 | C2×Q8 | D4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 3 | 3 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D14⋊6SD16 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 20 |
| 0 | 0 | 103 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 89 | 104 |
| 0 | 0 | 89 | 24 |
| 100 | 100 | 0 | 0 |
| 13 | 100 | 0 | 0 |
| 0 | 0 | 41 | 99 |
| 0 | 0 | 7 | 72 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,23,103,0,0,20,11],[1,0,0,0,0,1,0,0,0,0,89,89,0,0,104,24],[100,13,0,0,100,100,0,0,0,0,41,7,0,0,99,72],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112] >;
D14⋊6SD16 in GAP, Magma, Sage, TeX
D_{14}\rtimes_6{\rm SD}_{16} % in TeX
G:=Group("D14:6SD16"); // GroupNames label
G:=SmallGroup(448,703);
// by ID
G=gap.SmallGroup(448,703);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,184,851,438,102,18822]);
// 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^3>;
// generators/relations