metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊15D14, Q16⋊13D14, D28.45D4, SD16⋊11D14, D56⋊19C22, C28.16C24, C56.38C23, Dic14.45D4, D28.11C23, C4○D8⋊4D7, (D7×D8)⋊7C2, C7⋊3(D4○D8), C4○D4⋊1D14, (C2×C8)⋊13D14, C7⋊D4.1D4, C7⋊C8.7C23, (C2×D56)⋊23C2, D56⋊C2⋊6C2, D4⋊D7⋊3C22, C4.143(D4×D7), (D4×D7)⋊2C22, (C8×D7)⋊8C22, Q8⋊D7⋊2C22, D4⋊8D14⋊5C2, Q8.D14⋊7C2, D4⋊D14⋊7C2, C22.8(D4×D7), (C2×C56)⋊12C22, D14.29(C2×D4), C28.349(C2×D4), (C7×D8)⋊13C22, (C4×D7).9C23, C8.16(C22×D7), C4.16(C23×D7), D28.2C4⋊7C2, (C2×D28)⋊34C22, C8⋊D7⋊11C22, Dic7.34(C2×D4), Q8⋊2D7⋊2C22, (C7×Q16)⋊11C22, (C7×D4).10C23, D4.10(C22×D7), (C7×Q8).10C23, Q8.10(C22×D7), (C2×C28).533C23, (C7×SD16)⋊11C22, C4○D28.54C22, C14.117(C22×D4), C4.Dic7⋊30C22, C2.90(C2×D4×D7), (C7×C4○D8)⋊5C2, (C2×C14).13(C2×D4), (C7×C4○D4)⋊3C22, (C2×C4).232(C22×D7), SmallGroup(448,1222)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊15D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=a2b, dcd=c-1 >
Subgroups: 1620 in 268 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C4○D4, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×D8, C4○D8, C4○D8, C8⋊C22, 2+ 1+4, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, D28, D28, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, D4○D8, C8×D7, C8⋊D7, D56, C4.Dic7, D4⋊D7, Q8⋊D7, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, Q8⋊2D7, C7×C4○D4, D28.2C4, C2×D56, D7×D8, D56⋊C2, Q8.D14, D4⋊D14, C7×C4○D8, D4⋊8D14, D8⋊15D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○D8, D4×D7, C23×D7, C2×D4×D7, D8⋊15D14
►On 112 points
(1 16 40 54 65 109 71 92)(2 17 41 55 66 110 72 93)(3 18 42 56 67 111 73 94)(4 19 29 43 68 112 74 95)(5 20 30 44 69 99 75 96)(6 21 31 45 70 100 76 97)(7 22 32 46 57 101 77 98)(8 23 33 47 58 102 78 85)(9 24 34 48 59 103 79 86)(10 25 35 49 60 104 80 87)(11 26 36 50 61 105 81 88)(12 27 37 51 62 106 82 89)(13 28 38 52 63 107 83 90)(14 15 39 53 64 108 84 91)
(2 66)(4 68)(6 70)(8 58)(10 60)(12 62)(14 64)(15 53)(16 92)(17 55)(18 94)(19 43)(20 96)(21 45)(22 98)(23 47)(24 86)(25 49)(26 88)(27 51)(28 90)(30 75)(32 77)(34 79)(36 81)(38 83)(40 71)(42 73)(44 99)(46 101)(48 103)(50 105)(52 107)(54 109)(56 111)(85 102)(87 104)(89 106)(91 108)(93 110)(95 112)(97 100)
(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 46)(2 45)(3 44)(4 43)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 33)(16 32)(17 31)(18 30)(19 29)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(57 92)(58 91)(59 90)(60 89)(61 88)(62 87)(63 86)(64 85)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 101)(72 100)(73 99)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)
G:=sub<Sym(112)| (1,16,40,54,65,109,71,92)(2,17,41,55,66,110,72,93)(3,18,42,56,67,111,73,94)(4,19,29,43,68,112,74,95)(5,20,30,44,69,99,75,96)(6,21,31,45,70,100,76,97)(7,22,32,46,57,101,77,98)(8,23,33,47,58,102,78,85)(9,24,34,48,59,103,79,86)(10,25,35,49,60,104,80,87)(11,26,36,50,61,105,81,88)(12,27,37,51,62,106,82,89)(13,28,38,52,63,107,83,90)(14,15,39,53,64,108,84,91), (2,66)(4,68)(6,70)(8,58)(10,60)(12,62)(14,64)(15,53)(16,92)(17,55)(18,94)(19,43)(20,96)(21,45)(22,98)(23,47)(24,86)(25,49)(26,88)(27,51)(28,90)(30,75)(32,77)(34,79)(36,81)(38,83)(40,71)(42,73)(44,99)(46,101)(48,103)(50,105)(52,107)(54,109)(56,111)(85,102)(87,104)(89,106)(91,108)(93,110)(95,112)(97,100), (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), (1,46)(2,45)(3,44)(4,43)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,33)(16,32)(17,31)(18,30)(19,29)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,101)(72,100)(73,99)(74,112)(75,111)(76,110)(77,109)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102)>;
G:=Group( (1,16,40,54,65,109,71,92)(2,17,41,55,66,110,72,93)(3,18,42,56,67,111,73,94)(4,19,29,43,68,112,74,95)(5,20,30,44,69,99,75,96)(6,21,31,45,70,100,76,97)(7,22,32,46,57,101,77,98)(8,23,33,47,58,102,78,85)(9,24,34,48,59,103,79,86)(10,25,35,49,60,104,80,87)(11,26,36,50,61,105,81,88)(12,27,37,51,62,106,82,89)(13,28,38,52,63,107,83,90)(14,15,39,53,64,108,84,91), (2,66)(4,68)(6,70)(8,58)(10,60)(12,62)(14,64)(15,53)(16,92)(17,55)(18,94)(19,43)(20,96)(21,45)(22,98)(23,47)(24,86)(25,49)(26,88)(27,51)(28,90)(30,75)(32,77)(34,79)(36,81)(38,83)(40,71)(42,73)(44,99)(46,101)(48,103)(50,105)(52,107)(54,109)(56,111)(85,102)(87,104)(89,106)(91,108)(93,110)(95,112)(97,100), (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), (1,46)(2,45)(3,44)(4,43)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,33)(16,32)(17,31)(18,30)(19,29)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,101)(72,100)(73,99)(74,112)(75,111)(76,110)(77,109)(78,108)(79,107)(80,106)(81,105)(82,104)(83,103)(84,102) );
G=PermutationGroup([[(1,16,40,54,65,109,71,92),(2,17,41,55,66,110,72,93),(3,18,42,56,67,111,73,94),(4,19,29,43,68,112,74,95),(5,20,30,44,69,99,75,96),(6,21,31,45,70,100,76,97),(7,22,32,46,57,101,77,98),(8,23,33,47,58,102,78,85),(9,24,34,48,59,103,79,86),(10,25,35,49,60,104,80,87),(11,26,36,50,61,105,81,88),(12,27,37,51,62,106,82,89),(13,28,38,52,63,107,83,90),(14,15,39,53,64,108,84,91)], [(2,66),(4,68),(6,70),(8,58),(10,60),(12,62),(14,64),(15,53),(16,92),(17,55),(18,94),(19,43),(20,96),(21,45),(22,98),(23,47),(24,86),(25,49),(26,88),(27,51),(28,90),(30,75),(32,77),(34,79),(36,81),(38,83),(40,71),(42,73),(44,99),(46,101),(48,103),(50,105),(52,107),(54,109),(56,111),(85,102),(87,104),(89,106),(91,108),(93,110),(95,112),(97,100)], [(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)], [(1,46),(2,45),(3,44),(4,43),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,33),(16,32),(17,31),(18,30),(19,29),(20,42),(21,41),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(57,92),(58,91),(59,90),(60,89),(61,88),(62,87),(63,86),(64,85),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,101),(72,100),(73,99),(74,112),(75,111),(76,110),(77,109),(78,108),(79,107),(80,106),(81,105),(82,104),(83,103),(84,102)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 2 | 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 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D4○D8 | D4×D7 | D4×D7 | D8⋊15D14 |
| kernel | D8⋊15D14 | D28.2C4 | C2×D56 | D7×D8 | D56⋊C2 | Q8.D14 | D4⋊D14 | C7×C4○D8 | D4⋊8D14 | Dic14 | D28 | C7⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 3 | 3 | 3 | 6 | 3 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D8⋊15D14 ►in GL4(𝔽113) generated by
| 82 | 0 | 31 | 0 |
| 0 | 82 | 0 | 31 |
| 82 | 0 | 82 | 0 |
| 0 | 82 | 0 | 82 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 94 | 19 |
| 0 | 0 | 94 | 13 |
| 19 | 94 | 0 | 0 |
| 19 | 100 | 0 | 0 |
| 16 | 16 | 97 | 97 |
| 37 | 97 | 76 | 16 |
| 97 | 97 | 97 | 97 |
| 76 | 16 | 76 | 16 |
G:=sub<GL(4,GF(113))| [82,0,82,0,0,82,0,82,31,0,82,0,0,31,0,82],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[0,0,19,19,0,0,94,100,94,94,0,0,19,13,0,0],[16,37,97,76,16,97,97,16,97,76,97,76,97,16,97,16] >;
D8⋊15D14 in GAP, Magma, Sage, TeX
D_8\rtimes_{15}D_{14} % in TeX
G:=Group("D8:15D14"); // GroupNames label
G:=SmallGroup(448,1222);
// by ID
G=gap.SmallGroup(448,1222);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,570,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations