metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.6D14, C28.21D8, C56.34D4, D56⋊14C22, C56.24C23, Dic28⋊11C22, (C2×D8)⋊7D7, (C14×D8)⋊1C2, C7⋊D16⋊5C2, C7⋊C16⋊3C22, D8.D7⋊5C2, C7⋊4(C16⋊C22), C14.63(C2×D8), (C2×C14).42D8, (C2×C8).83D14, C8.2(C7⋊D4), D56⋊7C2⋊2C2, C28.C8⋊2C2, C4.17(D4⋊D7), (C2×C28).180D4, C28.160(C2×D4), (C7×D8).6C22, C8.30(C22×D7), (C2×C56).31C22, C22.10(D4⋊D7), C4.2(C2×C7⋊D4), C2.18(C2×D4⋊D7), (C2×C4).79(C7⋊D4), SmallGroup(448,681)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.D14
G = < a,b,c,d | a8=b2=1, c14=d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c13 >
Subgroups: 484 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, M5(2), D16, SD32, C2×D8, C4○D8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×D4, C22×C14, C16⋊C22, C7⋊C16, C56⋊C2, D56, Dic28, C2×C56, C7×D8, C7×D8, C4○D28, D4×C14, C28.C8, C7⋊D16, D8.D7, D56⋊7C2, C14×D8, D8.D14
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C7⋊D4, C22×D7, C16⋊C22, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7, D8.D14
►On 112 points
Generators in S112
(1 45 22 38 15 31 8 52)(2 46 23 39 16 32 9 53)(3 47 24 40 17 33 10 54)(4 48 25 41 18 34 11 55)(5 49 26 42 19 35 12 56)(6 50 27 43 20 36 13 29)(7 51 28 44 21 37 14 30)(57 89 64 96 71 103 78 110)(58 90 65 97 72 104 79 111)(59 91 66 98 73 105 80 112)(60 92 67 99 74 106 81 85)(61 93 68 100 75 107 82 86)(62 94 69 101 76 108 83 87)(63 95 70 102 77 109 84 88)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(29 43)(31 45)(33 47)(35 49)(37 51)(39 53)(41 55)(57 96)(58 111)(59 98)(60 85)(61 100)(62 87)(63 102)(64 89)(65 104)(66 91)(67 106)(68 93)(69 108)(70 95)(71 110)(72 97)(73 112)(74 99)(75 86)(76 101)(77 88)(78 103)(79 90)(80 105)(81 92)(82 107)(83 94)(84 109)
(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 65 15 79)(2 78 16 64)(3 63 17 77)(4 76 18 62)(5 61 19 75)(6 74 20 60)(7 59 21 73)(8 72 22 58)(9 57 23 71)(10 70 24 84)(11 83 25 69)(12 68 26 82)(13 81 27 67)(14 66 28 80)(29 106 43 92)(30 91 44 105)(31 104 45 90)(32 89 46 103)(33 102 47 88)(34 87 48 101)(35 100 49 86)(36 85 50 99)(37 98 51 112)(38 111 52 97)(39 96 53 110)(40 109 54 95)(41 94 55 108)(42 107 56 93)
G:=sub<Sym(112)| (1,45,22,38,15,31,8,52)(2,46,23,39,16,32,9,53)(3,47,24,40,17,33,10,54)(4,48,25,41,18,34,11,55)(5,49,26,42,19,35,12,56)(6,50,27,43,20,36,13,29)(7,51,28,44,21,37,14,30)(57,89,64,96,71,103,78,110)(58,90,65,97,72,104,79,111)(59,91,66,98,73,105,80,112)(60,92,67,99,74,106,81,85)(61,93,68,100,75,107,82,86)(62,94,69,101,76,108,83,87)(63,95,70,102,77,109,84,88), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(29,43)(31,45)(33,47)(35,49)(37,51)(39,53)(41,55)(57,96)(58,111)(59,98)(60,85)(61,100)(62,87)(63,102)(64,89)(65,104)(66,91)(67,106)(68,93)(69,108)(70,95)(71,110)(72,97)(73,112)(74,99)(75,86)(76,101)(77,88)(78,103)(79,90)(80,105)(81,92)(82,107)(83,94)(84,109), (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), (1,65,15,79)(2,78,16,64)(3,63,17,77)(4,76,18,62)(5,61,19,75)(6,74,20,60)(7,59,21,73)(8,72,22,58)(9,57,23,71)(10,70,24,84)(11,83,25,69)(12,68,26,82)(13,81,27,67)(14,66,28,80)(29,106,43,92)(30,91,44,105)(31,104,45,90)(32,89,46,103)(33,102,47,88)(34,87,48,101)(35,100,49,86)(36,85,50,99)(37,98,51,112)(38,111,52,97)(39,96,53,110)(40,109,54,95)(41,94,55,108)(42,107,56,93)>;
G:=Group( (1,45,22,38,15,31,8,52)(2,46,23,39,16,32,9,53)(3,47,24,40,17,33,10,54)(4,48,25,41,18,34,11,55)(5,49,26,42,19,35,12,56)(6,50,27,43,20,36,13,29)(7,51,28,44,21,37,14,30)(57,89,64,96,71,103,78,110)(58,90,65,97,72,104,79,111)(59,91,66,98,73,105,80,112)(60,92,67,99,74,106,81,85)(61,93,68,100,75,107,82,86)(62,94,69,101,76,108,83,87)(63,95,70,102,77,109,84,88), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(29,43)(31,45)(33,47)(35,49)(37,51)(39,53)(41,55)(57,96)(58,111)(59,98)(60,85)(61,100)(62,87)(63,102)(64,89)(65,104)(66,91)(67,106)(68,93)(69,108)(70,95)(71,110)(72,97)(73,112)(74,99)(75,86)(76,101)(77,88)(78,103)(79,90)(80,105)(81,92)(82,107)(83,94)(84,109), (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), (1,65,15,79)(2,78,16,64)(3,63,17,77)(4,76,18,62)(5,61,19,75)(6,74,20,60)(7,59,21,73)(8,72,22,58)(9,57,23,71)(10,70,24,84)(11,83,25,69)(12,68,26,82)(13,81,27,67)(14,66,28,80)(29,106,43,92)(30,91,44,105)(31,104,45,90)(32,89,46,103)(33,102,47,88)(34,87,48,101)(35,100,49,86)(36,85,50,99)(37,98,51,112)(38,111,52,97)(39,96,53,110)(40,109,54,95)(41,94,55,108)(42,107,56,93) );
G=PermutationGroup([[(1,45,22,38,15,31,8,52),(2,46,23,39,16,32,9,53),(3,47,24,40,17,33,10,54),(4,48,25,41,18,34,11,55),(5,49,26,42,19,35,12,56),(6,50,27,43,20,36,13,29),(7,51,28,44,21,37,14,30),(57,89,64,96,71,103,78,110),(58,90,65,97,72,104,79,111),(59,91,66,98,73,105,80,112),(60,92,67,99,74,106,81,85),(61,93,68,100,75,107,82,86),(62,94,69,101,76,108,83,87),(63,95,70,102,77,109,84,88)], [(1,8),(2,23),(3,10),(4,25),(5,12),(6,27),(7,14),(9,16),(11,18),(13,20),(15,22),(17,24),(19,26),(21,28),(29,43),(31,45),(33,47),(35,49),(37,51),(39,53),(41,55),(57,96),(58,111),(59,98),(60,85),(61,100),(62,87),(63,102),(64,89),(65,104),(66,91),(67,106),(68,93),(69,108),(70,95),(71,110),(72,97),(73,112),(74,99),(75,86),(76,101),(77,88),(78,103),(79,90),(80,105),(81,92),(82,107),(83,94),(84,109)], [(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)], [(1,65,15,79),(2,78,16,64),(3,63,17,77),(4,76,18,62),(5,61,19,75),(6,74,20,60),(7,59,21,73),(8,72,22,58),(9,57,23,71),(10,70,24,84),(11,83,25,69),(12,68,26,82),(13,81,27,67),(14,66,28,80),(29,106,43,92),(30,91,44,105),(31,104,45,90),(32,89,46,103),(33,102,47,88),(34,87,48,101),(35,100,49,86),(36,85,50,99),(37,98,51,112),(38,111,52,97),(39,96,53,110),(40,109,54,95),(41,94,55,108),(42,107,56,93)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 14A | ··· | 14I | 14J | ··· | 14U | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 8 | 56 | 2 | 2 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | ··· | 2 | 8 | ··· | 8 | 28 | 28 | 28 | 28 | 4 | ··· | 4 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | C7⋊D4 | C7⋊D4 | C16⋊C22 | D4⋊D7 | D4⋊D7 | D8.D14 |
| kernel | D8.D14 | C28.C8 | C7⋊D16 | D8.D7 | D56⋊7C2 | C14×D8 | C56 | C2×C28 | C2×D8 | C28 | C2×C14 | C2×C8 | D8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D8.D14 ►in GL4(𝔽113) generated by
| 0 | 46 | 0 | 0 |
| 27 | 51 | 0 | 0 |
| 86 | 31 | 82 | 82 |
| 0 | 31 | 31 | 82 |
| 112 | 86 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 19 | 112 | 82 | 82 |
| 46 | 112 | 82 | 31 |
| 64 | 33 | 0 | 0 |
| 12 | 49 | 0 | 0 |
| 83 | 64 | 0 | 83 |
| 18 | 64 | 30 | 0 |
| 30 | 0 | 0 | 94 |
| 48 | 0 | 30 | 30 |
| 83 | 64 | 0 | 83 |
| 95 | 0 | 0 | 83 |
G:=sub<GL(4,GF(113))| [0,27,86,0,46,51,31,31,0,0,82,31,0,0,82,82],[112,0,19,46,86,1,112,112,0,0,82,82,0,0,82,31],[64,12,83,18,33,49,64,64,0,0,0,30,0,0,83,0],[30,48,83,95,0,0,64,0,0,30,0,0,94,30,83,83] >;
D8.D14 in GAP, Magma, Sage, TeX
D_8.D_{14} % in TeX
G:=Group("D8.D14"); // GroupNames label
G:=SmallGroup(448,681);
// by ID
G=gap.SmallGroup(448,681);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,387,675,185,192,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^14=d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^13>;
// generators/relations