metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊10D14, Q16⋊9D14, SD16⋊10D14, D56⋊22C22, C28.15C24, C56.37C23, D28.10C23, Dic28⋊19C22, Dic14.10C23, C4○D8⋊3D7, C4○D4⋊8D14, (C2×C8)⋊12D14, C7⋊C8.6C23, D8⋊D7⋊7C2, D56⋊C2⋊7C2, (C2×C56)⋊5C22, (C4×D7).51D4, C4.222(D4×D7), (D4×D7)⋊7C22, D56⋊7C2⋊7C2, (Q8×D7)⋊8C22, C22.5(D4×D7), D4⋊D7⋊13C22, Q16⋊D7⋊7C2, D14.53(C2×D4), C28.381(C2×D4), SD16⋊D7⋊7C2, C4○D28⋊6C22, (C7×D8)⋊15C22, Q8⋊D7⋊12C22, (C7×D4).9C23, (C4×D7).8C23, D4.9(C22×D7), C4.15(C23×D7), C8.15(C22×D7), D4.8D14⋊2C2, (C7×Q8).9C23, Q8.9(C22×D7), C8⋊D7⋊15C22, C56⋊C2⋊16C22, C7⋊2(D8⋊C22), D4.D7⋊12C22, (C2×Dic7).80D4, Dic7.58(C2×D4), (C7×Q16)⋊13C22, C7⋊Q16⋊11C22, (C22×D7).42D4, (C2×C28).532C23, (C7×SD16)⋊10C22, D4⋊2D7.5C22, C14.116(C22×D4), Q8⋊2D7.5C22, C2.89(C2×D4×D7), (C7×C4○D8)⋊3C2, (D7×C4○D4)⋊2C2, (C2×C8⋊D7)⋊1C2, (C2×C7⋊C8)⋊16C22, (C2×C14).12(C2×D4), (C7×C4○D4)⋊2C22, (C2×C4×D7).159C22, (C2×C4).619(C22×D7), SmallGroup(448,1221)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1332 in 262 conjugacy classes, 99 normal (33 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23 [×3], D7 [×4], C14, C14 [×3], C2×C8, C2×C8, M4(2) [×4], D8, D8 [×3], SD16 [×2], SD16 [×6], Q16, Q16 [×3], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×2], C4○D4 [×10], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×2], D14 [×2], D14 [×7], C2×C14, C2×C14 [×2], C2×M4(2), C4○D8, C4○D8 [×3], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14 [×2], C4×D7 [×4], C4×D7 [×6], D28 [×2], D28 [×2], C2×Dic7, C2×Dic7 [×2], C7⋊D4 [×6], C2×C28, C2×C28 [×2], C7×D4 [×2], C7×D4 [×2], C7×Q8 [×2], C22×D7, C22×D7 [×2], D8⋊C22, C8⋊D7 [×4], C56⋊C2 [×2], D56, Dic28, C2×C7⋊C8, D4⋊D7 [×2], D4.D7 [×2], Q8⋊D7 [×2], C7⋊Q16 [×2], C2×C56, C7×D8, C7×SD16 [×2], C7×Q16, C2×C4×D7, C2×C4×D7 [×2], C4○D28 [×2], C4○D28 [×2], D4×D7 [×2], D4×D7 [×2], D4⋊2D7 [×2], D4⋊2D7 [×2], Q8×D7 [×2], Q8⋊2D7 [×2], C7×C4○D4 [×2], C2×C8⋊D7, D56⋊7C2, D8⋊D7 [×2], D56⋊C2 [×2], SD16⋊D7 [×2], Q16⋊D7 [×2], D4.8D14 [×2], C7×C4○D8, D7×C4○D4 [×2], D8⋊10D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D8⋊C22, D4×D7 [×2], C23×D7, C2×D4×D7, D8⋊10D14
Generators and relations
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, ac=ca, dad=a5, cbc-1=dbd=a4b, dcd=c-1 >
►On 112 points
(1 43 33 17 36 9 52 27)(2 44 34 18 37 10 53 28)(3 45 35 19 38 11 54 22)(4 46 29 20 39 12 55 23)(5 47 30 21 40 13 56 24)(6 48 31 15 41 14 50 25)(7 49 32 16 42 8 51 26)(57 78 90 111 64 71 97 104)(58 79 91 112 65 72 98 105)(59 80 92 99 66 73 85 106)(60 81 93 100 67 74 86 107)(61 82 94 101 68 75 87 108)(62 83 95 102 69 76 88 109)(63 84 96 103 70 77 89 110)
(1 104)(2 112)(3 106)(4 100)(5 108)(6 102)(7 110)(8 96)(9 90)(10 98)(11 92)(12 86)(13 94)(14 88)(15 62)(16 70)(17 64)(18 58)(19 66)(20 60)(21 68)(22 59)(23 67)(24 61)(25 69)(26 63)(27 57)(28 65)(29 81)(30 75)(31 83)(32 77)(33 71)(34 79)(35 73)(36 111)(37 105)(38 99)(39 107)(40 101)(41 109)(42 103)(43 97)(44 91)(45 85)(46 93)(47 87)(48 95)(49 89)(50 76)(51 84)(52 78)(53 72)(54 80)(55 74)(56 82)
(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 37)(2 36)(3 42)(4 41)(5 40)(6 39)(7 38)(8 11)(9 10)(12 14)(15 20)(16 19)(17 18)(22 26)(23 25)(27 28)(29 50)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(43 44)(45 49)(46 48)(57 58)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(71 79)(72 78)(73 77)(74 76)(80 84)(81 83)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 98)(99 103)(100 102)(104 112)(105 111)(106 110)(107 109)
G:=sub<Sym(112)| (1,43,33,17,36,9,52,27)(2,44,34,18,37,10,53,28)(3,45,35,19,38,11,54,22)(4,46,29,20,39,12,55,23)(5,47,30,21,40,13,56,24)(6,48,31,15,41,14,50,25)(7,49,32,16,42,8,51,26)(57,78,90,111,64,71,97,104)(58,79,91,112,65,72,98,105)(59,80,92,99,66,73,85,106)(60,81,93,100,67,74,86,107)(61,82,94,101,68,75,87,108)(62,83,95,102,69,76,88,109)(63,84,96,103,70,77,89,110), (1,104)(2,112)(3,106)(4,100)(5,108)(6,102)(7,110)(8,96)(9,90)(10,98)(11,92)(12,86)(13,94)(14,88)(15,62)(16,70)(17,64)(18,58)(19,66)(20,60)(21,68)(22,59)(23,67)(24,61)(25,69)(26,63)(27,57)(28,65)(29,81)(30,75)(31,83)(32,77)(33,71)(34,79)(35,73)(36,111)(37,105)(38,99)(39,107)(40,101)(41,109)(42,103)(43,97)(44,91)(45,85)(46,93)(47,87)(48,95)(49,89)(50,76)(51,84)(52,78)(53,72)(54,80)(55,74)(56,82), (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), (1,37)(2,36)(3,42)(4,41)(5,40)(6,39)(7,38)(8,11)(9,10)(12,14)(15,20)(16,19)(17,18)(22,26)(23,25)(27,28)(29,50)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(43,44)(45,49)(46,48)(57,58)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,98)(99,103)(100,102)(104,112)(105,111)(106,110)(107,109)>;
G:=Group( (1,43,33,17,36,9,52,27)(2,44,34,18,37,10,53,28)(3,45,35,19,38,11,54,22)(4,46,29,20,39,12,55,23)(5,47,30,21,40,13,56,24)(6,48,31,15,41,14,50,25)(7,49,32,16,42,8,51,26)(57,78,90,111,64,71,97,104)(58,79,91,112,65,72,98,105)(59,80,92,99,66,73,85,106)(60,81,93,100,67,74,86,107)(61,82,94,101,68,75,87,108)(62,83,95,102,69,76,88,109)(63,84,96,103,70,77,89,110), (1,104)(2,112)(3,106)(4,100)(5,108)(6,102)(7,110)(8,96)(9,90)(10,98)(11,92)(12,86)(13,94)(14,88)(15,62)(16,70)(17,64)(18,58)(19,66)(20,60)(21,68)(22,59)(23,67)(24,61)(25,69)(26,63)(27,57)(28,65)(29,81)(30,75)(31,83)(32,77)(33,71)(34,79)(35,73)(36,111)(37,105)(38,99)(39,107)(40,101)(41,109)(42,103)(43,97)(44,91)(45,85)(46,93)(47,87)(48,95)(49,89)(50,76)(51,84)(52,78)(53,72)(54,80)(55,74)(56,82), (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), (1,37)(2,36)(3,42)(4,41)(5,40)(6,39)(7,38)(8,11)(9,10)(12,14)(15,20)(16,19)(17,18)(22,26)(23,25)(27,28)(29,50)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(43,44)(45,49)(46,48)(57,58)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,98)(99,103)(100,102)(104,112)(105,111)(106,110)(107,109) );
G=PermutationGroup([(1,43,33,17,36,9,52,27),(2,44,34,18,37,10,53,28),(3,45,35,19,38,11,54,22),(4,46,29,20,39,12,55,23),(5,47,30,21,40,13,56,24),(6,48,31,15,41,14,50,25),(7,49,32,16,42,8,51,26),(57,78,90,111,64,71,97,104),(58,79,91,112,65,72,98,105),(59,80,92,99,66,73,85,106),(60,81,93,100,67,74,86,107),(61,82,94,101,68,75,87,108),(62,83,95,102,69,76,88,109),(63,84,96,103,70,77,89,110)], [(1,104),(2,112),(3,106),(4,100),(5,108),(6,102),(7,110),(8,96),(9,90),(10,98),(11,92),(12,86),(13,94),(14,88),(15,62),(16,70),(17,64),(18,58),(19,66),(20,60),(21,68),(22,59),(23,67),(24,61),(25,69),(26,63),(27,57),(28,65),(29,81),(30,75),(31,83),(32,77),(33,71),(34,79),(35,73),(36,111),(37,105),(38,99),(39,107),(40,101),(41,109),(42,103),(43,97),(44,91),(45,85),(46,93),(47,87),(48,95),(49,89),(50,76),(51,84),(52,78),(53,72),(54,80),(55,74),(56,82)], [(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)], [(1,37),(2,36),(3,42),(4,41),(5,40),(6,39),(7,38),(8,11),(9,10),(12,14),(15,20),(16,19),(17,18),(22,26),(23,25),(27,28),(29,50),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(43,44),(45,49),(46,48),(57,58),(59,70),(60,69),(61,68),(62,67),(63,66),(64,65),(71,79),(72,78),(73,77),(74,76),(80,84),(81,83),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(97,98),(99,103),(100,102),(104,112),(105,111),(106,110),(107,109)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 80 | 0 | 0 | 0 | 0 |
| 24 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 89 | 33 | 0 | 0 | 0 | 0 |
| 99 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,98,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1,0,0,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,24,0,0,0,0,80,89,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1],[89,99,0,0,0,0,33,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D8⋊C22 | D4×D7 | D4×D7 | D8⋊10D14 |
| kernel | D8⋊10D14 | C2×C8⋊D7 | D56⋊7C2 | D8⋊D7 | D56⋊C2 | SD16⋊D7 | Q16⋊D7 | D4.8D14 | C7×C4○D8 | D7×C4○D4 | C4×D7 | C2×Dic7 | C22×D7 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 3 | 6 | 3 | 6 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{10}D_{14} % in TeX
G:=Group("D8:10D14"); // GroupNames label
G:=SmallGroup(448,1221);
// by ID
G=gap.SmallGroup(448,1221);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations