metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊3D7, C8.8D14, D14.1D4, D4.1D14, Dic28⋊4C2, C56.6C22, C28.3C23, Dic7.12D4, Dic14.1C22, (C8×D7)⋊2C2, (C7×D8)⋊3C2, C7⋊2(C4○D8), D4.D7⋊2C2, C2.17(D4×D7), C7⋊C8.5C22, D4⋊2D7⋊2C2, C14.29(C2×D4), C4.3(C22×D7), (C4×D7).8C22, (C7×D4).1C22, SmallGroup(224,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊3D7
G = < a,b,c,d | a8=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 254 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, D4, Q8, D7, C14, C14, C2×C8, D8, SD16, Q16, C4○D4, Dic7, Dic7, C28, D14, C2×C14, C4○D8, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C8×D7, Dic28, D4.D7, C7×D8, D4⋊2D7, D8⋊3D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4○D8, C22×D7, D4×D7, D8⋊3D7
►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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 50)(10 49)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 42)(18 41)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 112)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 81)(64 88)(65 93)(66 92)(67 91)(68 90)(69 89)(70 96)(71 95)(72 94)(73 98)(74 97)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)
(1 102 14 89 107 47 57)(2 103 15 90 108 48 58)(3 104 16 91 109 41 59)(4 97 9 92 110 42 60)(5 98 10 93 111 43 61)(6 99 11 94 112 44 62)(7 100 12 95 105 45 63)(8 101 13 96 106 46 64)(17 84 29 74 50 66 34)(18 85 30 75 51 67 35)(19 86 31 76 52 68 36)(20 87 32 77 53 69 37)(21 88 25 78 54 70 38)(22 81 26 79 55 71 39)(23 82 27 80 56 72 40)(24 83 28 73 49 65 33)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 110)(10 111)(11 112)(12 105)(13 106)(14 107)(15 108)(16 109)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(25 84)(26 85)(27 86)(28 87)(29 88)(30 81)(31 82)(32 83)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(41 104)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(65 69)(66 70)(67 71)(68 72)
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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,50)(10,49)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,112)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,81)(64,88)(65,93)(66,92)(67,91)(68,90)(69,89)(70,96)(71,95)(72,94)(73,98)(74,97)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99), (1,102,14,89,107,47,57)(2,103,15,90,108,48,58)(3,104,16,91,109,41,59)(4,97,9,92,110,42,60)(5,98,10,93,111,43,61)(6,99,11,94,112,44,62)(7,100,12,95,105,45,63)(8,101,13,96,106,46,64)(17,84,29,74,50,66,34)(18,85,30,75,51,67,35)(19,86,31,76,52,68,36)(20,87,32,77,53,69,37)(21,88,25,78,54,70,38)(22,81,26,79,55,71,39)(23,82,27,80,56,72,40)(24,83,28,73,49,65,33), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,110)(10,111)(11,112)(12,105)(13,106)(14,107)(15,108)(16,109)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(41,104)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(65,69)(66,70)(67,71)(68,72)>;
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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,50)(10,49)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,112)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,81)(64,88)(65,93)(66,92)(67,91)(68,90)(69,89)(70,96)(71,95)(72,94)(73,98)(74,97)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99), (1,102,14,89,107,47,57)(2,103,15,90,108,48,58)(3,104,16,91,109,41,59)(4,97,9,92,110,42,60)(5,98,10,93,111,43,61)(6,99,11,94,112,44,62)(7,100,12,95,105,45,63)(8,101,13,96,106,46,64)(17,84,29,74,50,66,34)(18,85,30,75,51,67,35)(19,86,31,76,52,68,36)(20,87,32,77,53,69,37)(21,88,25,78,54,70,38)(22,81,26,79,55,71,39)(23,82,27,80,56,72,40)(24,83,28,73,49,65,33), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,110)(10,111)(11,112)(12,105)(13,106)(14,107)(15,108)(16,109)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(41,104)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(65,69)(66,70)(67,71)(68,72) );
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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,50),(10,49),(11,56),(12,55),(13,54),(14,53),(15,52),(16,51),(17,42),(18,41),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(33,111),(34,110),(35,109),(36,108),(37,107),(38,106),(39,105),(40,112),(57,87),(58,86),(59,85),(60,84),(61,83),(62,82),(63,81),(64,88),(65,93),(66,92),(67,91),(68,90),(69,89),(70,96),(71,95),(72,94),(73,98),(74,97),(75,104),(76,103),(77,102),(78,101),(79,100),(80,99)], [(1,102,14,89,107,47,57),(2,103,15,90,108,48,58),(3,104,16,91,109,41,59),(4,97,9,92,110,42,60),(5,98,10,93,111,43,61),(6,99,11,94,112,44,62),(7,100,12,95,105,45,63),(8,101,13,96,106,46,64),(17,84,29,74,50,66,34),(18,85,30,75,51,67,35),(19,86,31,76,52,68,36),(20,87,32,77,53,69,37),(21,88,25,78,54,70,38),(22,81,26,79,55,71,39),(23,82,27,80,56,72,40),(24,83,28,73,49,65,33)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,110),(10,111),(11,112),(12,105),(13,106),(14,107),(15,108),(16,109),(17,78),(18,79),(19,80),(20,73),(21,74),(22,75),(23,76),(24,77),(25,84),(26,85),(27,86),(28,87),(29,88),(30,81),(31,82),(32,83),(33,53),(34,54),(35,55),(36,56),(37,49),(38,50),(39,51),(40,52),(41,104),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(65,69),(66,70),(67,71),(68,72)]])
D8⋊3D7 is a maximal subgroup of
D8⋊D14 D16⋊3D7 SD32⋊D7 SD32⋊3D7 D8⋊13D14 D7×C4○D8 D8.10D14 SD16⋊D14 D8⋊6D14
D8⋊3D7 is a maximal quotient of
Dic7⋊6SD16 D4.2Dic14 Dic14.D4 (C8×Dic7)⋊C2 D4⋊2D7⋊C4 D14⋊SD16 D4.D28 C56⋊1C4⋊C2 Dic28⋊6C4 Dic14.2Q8 C56.4Q8 C8.27(C4×D7) D14⋊2Q16 C2.D8⋊7D7 D8×Dic7 (C2×D8).D7 C56.22D4 C56⋊6D4 Dic14⋊D4
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | ··· | 14I | 28A | 28B | 28C | 56A | ··· | 56F |
| 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 | 56 | ··· | 56 |
| size | 1 | 1 | 4 | 4 | 14 | 2 | 7 | 7 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C4○D8 | D4×D7 | D8⋊3D7 |
| kernel | D8⋊3D7 | C8×D7 | Dic28 | D4.D7 | C7×D8 | D4⋊2D7 | Dic7 | D14 | D8 | C8 | D4 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 4 | 3 | 6 |
Matrix representation of D8⋊3D7 ►in GL4(𝔽113) generated by
| 18 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 44 | 0 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 1 |
| 0 | 0 | 32 | 80 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 32 | 1 |
G:=sub<GL(4,GF(113))| [18,0,0,0,0,44,0,0,0,0,112,0,0,0,0,112],[0,18,0,0,44,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,32,0,0,1,80],[1,0,0,0,0,112,0,0,0,0,112,32,0,0,0,1] >;
D8⋊3D7 in GAP, Magma, Sage, TeX
D_8\rtimes_3D_7
% in TeX
G:=Group("D8:3D7"); // GroupNames label
G:=SmallGroup(224,107);
// by ID
G=gap.SmallGroup(224,107);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,362,116,297,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations