metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D14, C28.57D4, Q8.8D14, C28.18C23, D28.12C22, Dic14.11C22, D4⋊D7⋊7C2, C4○D4⋊2D7, C7⋊5(C4○D8), Q8⋊D7⋊7C2, C4○D28⋊4C2, D4.D7⋊7C2, C7⋊Q16⋊7C2, (C2×C14).9D4, (C2×C4).59D14, C14.60(C2×D4), C7⋊C8.10C22, C4.32(C7⋊D4), (C7×D4).8C22, C4.18(C22×D7), (C7×Q8).8C22, (C2×C28).43C22, C22.1(C7⋊D4), (C2×C7⋊C8)⋊8C2, (C7×C4○D4)⋊2C2, C2.24(C2×C7⋊D4), SmallGroup(224,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.8D14
G = < a,b,c,d | a4=b2=1, c14=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c13 >
Subgroups: 230 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4○D8, C7⋊C8, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C4○D28, C7×C4○D4, D4.8D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4○D8, C7⋊D4, C22×D7, C2×C7⋊D4, D4.8D14
►On 112 points
(1 71 15 57)(2 72 16 58)(3 73 17 59)(4 74 18 60)(5 75 19 61)(6 76 20 62)(7 77 21 63)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 98 43 112)(30 99 44 85)(31 100 45 86)(32 101 46 87)(33 102 47 88)(34 103 48 89)(35 104 49 90)(36 105 50 91)(37 106 51 92)(38 107 52 93)(39 108 53 94)(40 109 54 95)(41 110 55 96)(42 111 56 97)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)
(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 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 87 43 101)(30 100 44 86)(31 85 45 99)(32 98 46 112)(33 111 47 97)(34 96 48 110)(35 109 49 95)(36 94 50 108)(37 107 51 93)(38 92 52 106)(39 105 53 91)(40 90 54 104)(41 103 55 89)(42 88 56 102)(57 84 71 70)(58 69 72 83)(59 82 73 68)(60 67 74 81)(61 80 75 66)(62 65 76 79)(63 78 77 64)
G:=sub<Sym(112)| (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,98,43,112)(30,99,44,85)(31,100,45,86)(32,101,46,87)(33,102,47,88)(34,103,48,89)(35,104,49,90)(36,105,50,91)(37,106,51,92)(38,107,52,93)(39,108,53,94)(40,109,54,95)(41,110,55,96)(42,111,56,97), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,87,43,101)(30,100,44,86)(31,85,45,99)(32,98,46,112)(33,111,47,97)(34,96,48,110)(35,109,49,95)(36,94,50,108)(37,107,51,93)(38,92,52,106)(39,105,53,91)(40,90,54,104)(41,103,55,89)(42,88,56,102)(57,84,71,70)(58,69,72,83)(59,82,73,68)(60,67,74,81)(61,80,75,66)(62,65,76,79)(63,78,77,64)>;
G:=Group( (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,98,43,112)(30,99,44,85)(31,100,45,86)(32,101,46,87)(33,102,47,88)(34,103,48,89)(35,104,49,90)(36,105,50,91)(37,106,51,92)(38,107,52,93)(39,108,53,94)(40,109,54,95)(41,110,55,96)(42,111,56,97), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68), (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), (1,14,15,28)(2,27,16,13)(3,12,17,26)(4,25,18,11)(5,10,19,24)(6,23,20,9)(7,8,21,22)(29,87,43,101)(30,100,44,86)(31,85,45,99)(32,98,46,112)(33,111,47,97)(34,96,48,110)(35,109,49,95)(36,94,50,108)(37,107,51,93)(38,92,52,106)(39,105,53,91)(40,90,54,104)(41,103,55,89)(42,88,56,102)(57,84,71,70)(58,69,72,83)(59,82,73,68)(60,67,74,81)(61,80,75,66)(62,65,76,79)(63,78,77,64) );
G=PermutationGroup([[(1,71,15,57),(2,72,16,58),(3,73,17,59),(4,74,18,60),(5,75,19,61),(6,76,20,62),(7,77,21,63),(8,78,22,64),(9,79,23,65),(10,80,24,66),(11,81,25,67),(12,82,26,68),(13,83,27,69),(14,84,28,70),(29,98,43,112),(30,99,44,85),(31,100,45,86),(32,101,46,87),(33,102,47,88),(34,103,48,89),(35,104,49,90),(36,105,50,91),(37,106,51,92),(38,107,52,93),(39,108,53,94),(40,109,54,95),(41,110,55,96),(42,111,56,97)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,57),(46,58),(47,59),(48,60),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68)], [(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)], [(1,14,15,28),(2,27,16,13),(3,12,17,26),(4,25,18,11),(5,10,19,24),(6,23,20,9),(7,8,21,22),(29,87,43,101),(30,100,44,86),(31,85,45,99),(32,98,46,112),(33,111,47,97),(34,96,48,110),(35,109,49,95),(36,94,50,108),(37,107,51,93),(38,92,52,106),(39,105,53,91),(40,90,54,104),(41,103,55,89),(42,88,56,102),(57,84,71,70),(58,69,72,83),(59,82,73,68),(60,67,74,81),(61,80,75,66),(62,65,76,79),(63,78,77,64)]])
D4.8D14 is a maximal subgroup of
M4(2).22D14 C42.196D14 C56.93D4 C56.50D4 D7×C4○D8 D8⋊10D14 D8⋊5D14 D8⋊6D14 C56.C23 D28.44D4 C28.C24 D28.32C23 D28.33C23 D28.34C23 D28.35C23
D4.8D14 is a maximal quotient of
C4⋊C4.233D14 C28.45(C4⋊C4) C4.(C2×D28) C4⋊C4.236D14 D4.3Dic14 C4×D4⋊D7 D4.1D28 C4×D4.D7 Q8.3Dic14 C4×Q8⋊D7 Q8.1D28 C4×C7⋊Q16 (C2×D4).D14 D28⋊17D4 C7⋊C8⋊22D4 C7⋊C8⋊23D4 C14.(C4○D8) D28.37D4 C7⋊C8⋊24D4 C7⋊C8.29D4 C42.61D14 C42.213D14 D28.23D4 C42.214D14 Dic14.4Q8 C42.215D14 D28.4Q8 C42.216D14 C28.(C2×D4) (C7×D4)⋊14D4 (C7×D4).32D4
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 28 | 1 | 1 | 2 | 4 | 28 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 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 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C7⋊D4 | D4.8D14 |
| kernel | D4.8D14 | C2×C7⋊C8 | D4⋊D7 | D4.D7 | Q8⋊D7 | C7⋊Q16 | C4○D28 | C7×C4○D4 | C28 | C2×C14 | C4○D4 | C2×C4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 |
Matrix representation of D4.8D14 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 106 |
| 0 | 0 | 81 | 112 |
| 91 | 74 | 0 | 0 |
| 108 | 22 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 88 | 0 |
| 88 | 54 | 0 | 0 |
| 33 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 88 | 54 | 0 | 0 |
| 1 | 25 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 28 | 15 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,81,0,0,106,112],[91,108,0,0,74,22,0,0,0,0,0,88,0,0,9,0],[88,33,0,0,54,1,0,0,0,0,98,0,0,0,0,98],[88,1,0,0,54,25,0,0,0,0,98,28,0,0,0,15] >;
D4.8D14 in GAP, Magma, Sage, TeX
D_4._8D_{14} % in TeX
G:=Group("D4.8D14"); // GroupNames label
G:=SmallGroup(224,145);
// by ID
G=gap.SmallGroup(224,145);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,579,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^14=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^13>;
// generators/relations