metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊11D14, Q16⋊10D14, D28.46D4, SD16⋊15D14, C56.43C23, C28.17C24, Dic14.46D4, D28.12C23, Dic14.11C23, C4○D8⋊5D7, C4○D4⋊2D14, (C2×C8)⋊14D14, C7⋊D4.2D4, C7⋊C8.8C23, D8⋊D7⋊6C2, D4⋊D7⋊4C22, (D7×SD16)⋊6C2, C7⋊3(D4○SD16), C4.144(D4×D7), Q8⋊D7⋊3C22, D4⋊D14⋊8C2, D4⋊8D14⋊6C2, (Q8×D7)⋊2C22, C22.9(D4×D7), (C2×C56)⋊17C22, Q16⋊D7⋊6C2, D14.30(C2×D4), C28.350(C2×D4), (C8×D7)⋊10C22, (C7×D8)⋊16C22, D4.D7⋊3C22, (D4×D7).2C22, C7⋊Q16⋊2C22, C4.17(C23×D7), C8.17(C22×D7), SD16⋊3D7⋊6C2, D4.9D14⋊7C2, D4⋊2D7⋊2C22, C56⋊C2⋊21C22, C8⋊D7⋊16C22, Dic7.35(C2×D4), (C7×Q16)⋊14C22, (C7×D4).11C23, (C4×D7).10C23, D4.11(C22×D7), D4.10D14⋊5C2, D28.2C4⋊10C2, (C7×Q8).11C23, Q8.11(C22×D7), (C2×C28).534C23, (C7×SD16)⋊16C22, C4○D28.55C22, C14.118(C22×D4), C4.Dic7⋊31C22, Q8⋊2D7.2C22, (C2×Dic14)⋊38C22, (C2×D28).181C22, C2.91(C2×D4×D7), (C7×C4○D8)⋊7C2, (C2×C56⋊C2)⋊27C2, (C2×C14).14(C2×D4), (C7×C4○D4)⋊4C22, (C2×C4).233(C22×D7), SmallGroup(448,1223)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1364 in 258 conjugacy classes, 99 normal (53 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×14], Q8 [×2], Q8 [×6], C23 [×3], D7 [×4], C14, C14 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, D8 [×2], SD16 [×2], SD16 [×8], Q16, Q16 [×2], C2×D4 [×6], C2×Q8 [×4], C4○D4 [×2], C4○D4 [×9], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×2], D14 [×2], D14 [×5], C2×C14, C2×C14 [×2], C8○D4, C2×SD16 [×3], C4○D8, C4○D8 [×2], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C7⋊C8 [×2], C56 [×2], Dic14, Dic14 [×2], Dic14 [×3], C4×D7 [×2], C4×D7 [×4], D28, D28 [×2], D28 [×3], C2×Dic7 [×3], C7⋊D4 [×2], C7⋊D4 [×4], C2×C28, C2×C28 [×2], C7×D4 [×2], C7×D4 [×2], C7×Q8 [×2], C22×D7 [×3], D4○SD16, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×4], C4.Dic7, D4⋊D7 [×2], D4.D7 [×2], Q8⋊D7 [×2], C7⋊Q16 [×2], C2×C56, C7×D8, C7×SD16 [×2], C7×Q16, C2×Dic14, C2×Dic14, C2×D28, C2×D28, C4○D28, 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], D28.2C4, C2×C56⋊C2, D8⋊D7 [×2], D7×SD16 [×2], SD16⋊3D7 [×2], Q16⋊D7 [×2], D4⋊D14, D4.9D14, C7×C4○D8, D4⋊8D14, D4.10D14, D8⋊11D14
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⋊11D14
Generators and relations
G = < a,b,c,d | a8=b2=c14=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a4b, dbd=a6b, dcd=c-1 >
►On 112 points
(1 16 106 80 87 50 60 35)(2 17 107 81 88 51 61 36)(3 18 108 82 89 52 62 37)(4 19 109 83 90 53 63 38)(5 20 110 84 91 54 64 39)(6 21 111 71 92 55 65 40)(7 22 112 72 93 56 66 41)(8 23 99 73 94 43 67 42)(9 24 100 74 95 44 68 29)(10 25 101 75 96 45 69 30)(11 26 102 76 97 46 70 31)(12 27 103 77 98 47 57 32)(13 28 104 78 85 48 58 33)(14 15 105 79 86 49 59 34)
(1 35)(2 81)(3 37)(4 83)(5 39)(6 71)(7 41)(8 73)(9 29)(10 75)(11 31)(12 77)(13 33)(14 79)(15 105)(16 60)(17 107)(18 62)(19 109)(20 64)(21 111)(22 66)(23 99)(24 68)(25 101)(26 70)(27 103)(28 58)(30 96)(32 98)(34 86)(36 88)(38 90)(40 92)(42 94)(43 67)(44 100)(45 69)(46 102)(47 57)(48 104)(49 59)(50 106)(51 61)(52 108)(53 63)(54 110)(55 65)(56 112)(72 93)(74 95)(76 97)(78 85)(80 87)(82 89)(84 91)
(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)
(1 112)(2 111)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 100)(14 99)(15 43)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 33)(30 32)(34 42)(35 41)(36 40)(37 39)(57 96)(58 95)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 98)(70 97)(71 81)(72 80)(73 79)(74 78)(75 77)(82 84)
G:=sub<Sym(112)| (1,16,106,80,87,50,60,35)(2,17,107,81,88,51,61,36)(3,18,108,82,89,52,62,37)(4,19,109,83,90,53,63,38)(5,20,110,84,91,54,64,39)(6,21,111,71,92,55,65,40)(7,22,112,72,93,56,66,41)(8,23,99,73,94,43,67,42)(9,24,100,74,95,44,68,29)(10,25,101,75,96,45,69,30)(11,26,102,76,97,46,70,31)(12,27,103,77,98,47,57,32)(13,28,104,78,85,48,58,33)(14,15,105,79,86,49,59,34), (1,35)(2,81)(3,37)(4,83)(5,39)(6,71)(7,41)(8,73)(9,29)(10,75)(11,31)(12,77)(13,33)(14,79)(15,105)(16,60)(17,107)(18,62)(19,109)(20,64)(21,111)(22,66)(23,99)(24,68)(25,101)(26,70)(27,103)(28,58)(30,96)(32,98)(34,86)(36,88)(38,90)(40,92)(42,94)(43,67)(44,100)(45,69)(46,102)(47,57)(48,104)(49,59)(50,106)(51,61)(52,108)(53,63)(54,110)(55,65)(56,112)(72,93)(74,95)(76,97)(78,85)(80,87)(82,89)(84,91), (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), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,43)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,33)(30,32)(34,42)(35,41)(36,40)(37,39)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,98)(70,97)(71,81)(72,80)(73,79)(74,78)(75,77)(82,84)>;
G:=Group( (1,16,106,80,87,50,60,35)(2,17,107,81,88,51,61,36)(3,18,108,82,89,52,62,37)(4,19,109,83,90,53,63,38)(5,20,110,84,91,54,64,39)(6,21,111,71,92,55,65,40)(7,22,112,72,93,56,66,41)(8,23,99,73,94,43,67,42)(9,24,100,74,95,44,68,29)(10,25,101,75,96,45,69,30)(11,26,102,76,97,46,70,31)(12,27,103,77,98,47,57,32)(13,28,104,78,85,48,58,33)(14,15,105,79,86,49,59,34), (1,35)(2,81)(3,37)(4,83)(5,39)(6,71)(7,41)(8,73)(9,29)(10,75)(11,31)(12,77)(13,33)(14,79)(15,105)(16,60)(17,107)(18,62)(19,109)(20,64)(21,111)(22,66)(23,99)(24,68)(25,101)(26,70)(27,103)(28,58)(30,96)(32,98)(34,86)(36,88)(38,90)(40,92)(42,94)(43,67)(44,100)(45,69)(46,102)(47,57)(48,104)(49,59)(50,106)(51,61)(52,108)(53,63)(54,110)(55,65)(56,112)(72,93)(74,95)(76,97)(78,85)(80,87)(82,89)(84,91), (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), (1,112)(2,111)(3,110)(4,109)(5,108)(6,107)(7,106)(8,105)(9,104)(10,103)(11,102)(12,101)(13,100)(14,99)(15,43)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,33)(30,32)(34,42)(35,41)(36,40)(37,39)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,98)(70,97)(71,81)(72,80)(73,79)(74,78)(75,77)(82,84) );
G=PermutationGroup([(1,16,106,80,87,50,60,35),(2,17,107,81,88,51,61,36),(3,18,108,82,89,52,62,37),(4,19,109,83,90,53,63,38),(5,20,110,84,91,54,64,39),(6,21,111,71,92,55,65,40),(7,22,112,72,93,56,66,41),(8,23,99,73,94,43,67,42),(9,24,100,74,95,44,68,29),(10,25,101,75,96,45,69,30),(11,26,102,76,97,46,70,31),(12,27,103,77,98,47,57,32),(13,28,104,78,85,48,58,33),(14,15,105,79,86,49,59,34)], [(1,35),(2,81),(3,37),(4,83),(5,39),(6,71),(7,41),(8,73),(9,29),(10,75),(11,31),(12,77),(13,33),(14,79),(15,105),(16,60),(17,107),(18,62),(19,109),(20,64),(21,111),(22,66),(23,99),(24,68),(25,101),(26,70),(27,103),(28,58),(30,96),(32,98),(34,86),(36,88),(38,90),(40,92),(42,94),(43,67),(44,100),(45,69),(46,102),(47,57),(48,104),(49,59),(50,106),(51,61),(52,108),(53,63),(54,110),(55,65),(56,112),(72,93),(74,95),(76,97),(78,85),(80,87),(82,89),(84,91)], [(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)], [(1,112),(2,111),(3,110),(4,109),(5,108),(6,107),(7,106),(8,105),(9,104),(10,103),(11,102),(12,101),(13,100),(14,99),(15,43),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,33),(30,32),(34,42),(35,41),(36,40),(37,39),(57,96),(58,95),(59,94),(60,93),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,98),(70,97),(71,81),(72,80),(73,79),(74,78),(75,77),(82,84)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 76 | 71 | 37 | 42 |
| 42 | 37 | 71 | 76 |
| 76 | 71 | 76 | 71 |
| 42 | 37 | 42 | 37 |
| 76 | 71 | 37 | 42 |
| 42 | 37 | 71 | 76 |
| 37 | 42 | 37 | 42 |
| 71 | 76 | 71 | 76 |
| 0 | 0 | 4 | 109 |
| 0 | 0 | 4 | 81 |
| 109 | 4 | 0 | 0 |
| 109 | 32 | 0 | 0 |
| 0 | 0 | 33 | 33 |
| 0 | 0 | 104 | 80 |
| 33 | 33 | 0 | 0 |
| 104 | 80 | 0 | 0 |
G:=sub<GL(4,GF(113))| [76,42,76,42,71,37,71,37,37,71,76,42,42,76,71,37],[76,42,37,71,71,37,42,76,37,71,37,71,42,76,42,76],[0,0,109,109,0,0,4,32,4,4,0,0,109,81,0,0],[0,0,33,104,0,0,33,80,33,104,0,0,33,80,0,0] >;
64 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 | ··· | 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 | 7 | 7 | 7 | 8 | 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 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D4○SD16 | D4×D7 | D4×D7 | D8⋊11D14 |
| kernel | D8⋊11D14 | D28.2C4 | C2×C56⋊C2 | D8⋊D7 | D7×SD16 | SD16⋊3D7 | Q16⋊D7 | D4⋊D14 | D4.9D14 | C7×C4○D8 | D4⋊8D14 | D4.10D14 | Dic14 | D28 | C7⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 6 | 3 | 6 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{11}D_{14} % in TeX
G:=Group("D8:11D14"); // GroupNames label
G:=SmallGroup(448,1223);
// by ID
G=gap.SmallGroup(448,1223);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,570,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^4*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations