metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊6D14, SD16⋊4D14, D28.42D4, C56.3C23, C28.22C24, M4(2)⋊10D14, Dic14.42D4, Dic28⋊2C22, D28.15C23, Dic14.15C23, C4○D4⋊4D14, C8⋊C22⋊5D7, C7⋊D4.5D4, D8⋊D7⋊4C2, D8⋊3D7⋊2C2, D28.C4⋊2C2, (D7×SD16)⋊2C2, C4.116(D4×D7), C7⋊4(D4○SD16), (C7×D8)⋊4C22, (C8×D7)⋊4C22, C7⋊C8.26C23, D4⋊6D14⋊8C2, (Q8×D7)⋊3C22, C8.3(C22×D7), D4⋊D7⋊15C22, D14.33(C2×D4), C28.243(C2×D4), C8.D14⋊2C2, SD16⋊D7⋊2C2, C8⋊D7⋊4C22, C56⋊C2⋊4C22, Q8⋊D7⋊14C22, (D4×D7).3C22, C4.22(C23×D7), C22.15(D4×D7), (C2×D4).117D14, D4.8D14⋊4C2, D4⋊2D7⋊4C22, D4.D7⋊14C22, Dic7.38(C2×D4), (C7×SD16)⋊4C22, C7⋊Q16⋊13C22, (C7×D4).15C23, (C4×D7).14C23, D4.15(C22×D7), D4.10D14⋊7C2, (C7×Q8).15C23, Q8.15(C22×D7), (C2×C28).113C23, C4○D28.29C22, C14.123(C22×D4), (C7×M4(2))⋊4C22, (C2×Dic14)⋊40C22, (D4×C14).168C22, C2.96(C2×D4×D7), (C7×C8⋊C22)⋊4C2, (C2×C7⋊C8)⋊18C22, (C2×D4.D7)⋊29C2, (C2×C14).68(C2×D4), (C7×C4○D4)⋊7C22, (C2×C4).97(C22×D7), SmallGroup(448,1228)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1292 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×2], D4 [×13], Q8, Q8 [×7], C23 [×3], D7 [×3], C14, C14 [×4], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×2], D8, SD16 [×2], SD16 [×8], Q16 [×3], C2×D4, C2×D4 [×5], C2×Q8 [×4], C4○D4, C4○D4 [×10], Dic7 [×2], Dic7 [×3], C28 [×2], C28, D14 [×2], D14 [×3], C2×C14, C2×C14 [×4], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22, C8⋊C22 [×2], C8.C22 [×3], 2+ (1+4), 2- (1+4), C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14 [×2], Dic14 [×3], C4×D7 [×2], C4×D7 [×3], D28 [×2], C2×Dic7 [×5], C7⋊D4 [×2], C7⋊D4 [×7], C2×C28, C2×C28, C7×D4, C7×D4 [×2], C7×D4 [×2], C7×Q8, C22×D7 [×2], C22×C14, D4○SD16, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], Dic28 [×2], C2×C7⋊C8, D4⋊D7, D4.D7, D4.D7 [×4], Q8⋊D7, C7⋊Q16, C7×M4(2), C7×D8 [×2], C7×SD16 [×2], C2×Dic14, C2×Dic14, C4○D28 [×2], C4○D28, D4×D7 [×2], D4×D7, D4⋊2D7 [×4], D4⋊2D7 [×3], Q8×D7 [×2], C2×C7⋊D4 [×2], D4×C14, C7×C4○D4, D28.C4, C8.D14, D8⋊D7 [×2], D8⋊3D7 [×2], D7×SD16 [×2], SD16⋊D7 [×2], C2×D4.D7, D4.8D14, C7×C8⋊C22, D4⋊6D14, D4.10D14, D8⋊6D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D4○SD16, D4×D7 [×2], C23×D7, C2×D4×D7, D8⋊6D14
Generators and relations
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >
►On 112 points
(1 46 104 30 94 67 21 80)(2 31 22 47 95 81 105 68)(3 48 106 32 96 69 23 82)(4 33 24 49 97 83 107 70)(5 50 108 34 98 57 25 84)(6 35 26 51 85 71 109 58)(7 52 110 36 86 59 27 72)(8 37 28 53 87 73 111 60)(9 54 112 38 88 61 15 74)(10 39 16 55 89 75 99 62)(11 56 100 40 90 63 17 76)(12 41 18 43 91 77 101 64)(13 44 102 42 92 65 19 78)(14 29 20 45 93 79 103 66)
(15 112)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(29 66)(30 67)(31 68)(32 69)(33 70)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)
(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)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 100)(12 99)(13 112)(14 111)(15 92)(16 91)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 98)(24 97)(25 96)(26 95)(27 94)(28 93)(29 37)(30 36)(31 35)(32 34)(38 42)(39 41)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(71 81)(72 80)(73 79)(74 78)(75 77)(82 84)
G:=sub<Sym(112)| (1,46,104,30,94,67,21,80)(2,31,22,47,95,81,105,68)(3,48,106,32,96,69,23,82)(4,33,24,49,97,83,107,70)(5,50,108,34,98,57,25,84)(6,35,26,51,85,71,109,58)(7,52,110,36,86,59,27,72)(8,37,28,53,87,73,111,60)(9,54,112,38,88,61,15,74)(10,39,16,55,89,75,99,62)(11,56,100,40,90,63,17,76)(12,41,18,43,91,77,101,64)(13,44,102,42,92,65,19,78)(14,29,20,45,93,79,103,66), (15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,112)(14,111)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,37)(30,36)(31,35)(32,34)(38,42)(39,41)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(71,81)(72,80)(73,79)(74,78)(75,77)(82,84)>;
G:=Group( (1,46,104,30,94,67,21,80)(2,31,22,47,95,81,105,68)(3,48,106,32,96,69,23,82)(4,33,24,49,97,83,107,70)(5,50,108,34,98,57,25,84)(6,35,26,51,85,71,109,58)(7,52,110,36,86,59,27,72)(8,37,28,53,87,73,111,60)(9,54,112,38,88,61,15,74)(10,39,16,55,89,75,99,62)(11,56,100,40,90,63,17,76)(12,41,18,43,91,77,101,64)(13,44,102,42,92,65,19,78)(14,29,20,45,93,79,103,66), (15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (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), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,112)(14,111)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,37)(30,36)(31,35)(32,34)(38,42)(39,41)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(71,81)(72,80)(73,79)(74,78)(75,77)(82,84) );
G=PermutationGroup([(1,46,104,30,94,67,21,80),(2,31,22,47,95,81,105,68),(3,48,106,32,96,69,23,82),(4,33,24,49,97,83,107,70),(5,50,108,34,98,57,25,84),(6,35,26,51,85,71,109,58),(7,52,110,36,86,59,27,72),(8,37,28,53,87,73,111,60),(9,54,112,38,88,61,15,74),(10,39,16,55,89,75,99,62),(11,56,100,40,90,63,17,76),(12,41,18,43,91,77,101,64),(13,44,102,42,92,65,19,78),(14,29,20,45,93,79,103,66)], [(15,112),(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,109),(27,110),(28,111),(29,66),(30,67),(31,68),(32,69),(33,70),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76)], [(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)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,100),(12,99),(13,112),(14,111),(15,92),(16,91),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,98),(24,97),(25,96),(26,95),(27,94),(28,93),(29,37),(30,36),(31,35),(32,34),(38,42),(39,41),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,70),(50,69),(51,68),(52,67),(53,66),(54,65),(55,64),(56,63),(71,81),(72,80),(73,79),(74,78),(75,77),(82,84)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 91 | 22 |
| 0 | 0 | 36 | 0 | 26 | 0 |
| 0 | 0 | 77 | 13 | 100 | 13 |
| 0 | 0 | 0 | 100 | 100 | 13 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 81 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 32 | 0 | 0 | 112 |
| 23 | 10 | 0 | 0 | 0 | 0 |
| 93 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 106 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 103 | 1 | 0 | 0 | 0 | 0 |
| 14 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 112 |
| 0 | 0 | 0 | 112 | 112 | 0 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,77,0,0,0,0,0,13,100,0,0,91,26,100,100,0,0,22,0,13,13],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,81,0,32,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[23,93,0,0,0,0,10,11,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,106,112,1,1,0,0,0,1,0,0],[103,14,0,0,0,0,1,10,0,0,0,0,0,0,112,0,0,0,0,0,7,1,112,112,0,0,0,0,0,112,0,0,0,0,112,0] >;
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14O | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D4○SD16 | D4×D7 | D4×D7 | D8⋊6D14 |
| kernel | D8⋊6D14 | D28.C4 | C8.D14 | D8⋊D7 | D8⋊3D7 | D7×SD16 | SD16⋊D7 | C2×D4.D7 | D4.8D14 | C7×C8⋊C22 | D4⋊6D14 | D4.10D14 | Dic14 | D28 | C7⋊D4 | C8⋊C22 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
In GAP, Magma, Sage, TeX
D_8\rtimes_6D_{14} % in TeX
G:=Group("D8:6D14"); // GroupNames label
G:=SmallGroup(448,1228);
// by ID
G=gap.SmallGroup(448,1228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,570,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations